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>nnex
IQB

O'BRIEN'S

MATHEMATICAL    TRACTS,

PART    I.

y^s  v&)C;".X£  .£L>

I  1.  -

MATHEMATICAL  TRACTS,

PART    I.

MATHEMATICAL   TRACTS,

PART   I.

LAPLACE'S    COEFFICIENTS,
THE    FIGURE    OF   THE    EARTH,

THE    MOTION    OF  A   RIGID    BODY  ABOUT   ITS
CENTER  OF  GRAVITY,

AND

PRECESSION  AND  NUTATION.

MATTHEW    O'BRIEN,    B.A.,

•MATHEMATICAL  LECTURER  OP  CAIUS  COLLEGE.

CAMBRIDGE:

PRINTED   AT    THE    UNIVERSITY   PRESS,

FOR  J.   &  J.   J.    DEIGHTON,   TRINITY  STREET;

AND

JOHN  W.  PARKER,  LONDON.

M.DCCC.XL.

Stack
Annex

233

PREFACE.

THE  subjects  treated  of  in  the  following  Tracts
are,  Laplace's  Coefficients;  the  Investigation  of  the
Figure  of  the  Earth  on  the  Hypothesis  of  its  Original
Fluidity ;  the  Equations  of  Motion  of  a  Rigid  Body
about  its  Center  of  Gravity;  and  the  Application  of
these  Equations  to  the  case  of  the  Earth.  The
first  of  these  subjects  should  be  familiar  to  every
Mathematical  Student,  both  for  its  own  sake,  and
also  on  account  of  the  many  branches  of  Physical
Science  to  which  it  is  applicable.  The  second  sub-
ject is  extremely  interesting  as  a  physical  theory,
bearing  upon  the  original  state  of  the  Earth  and
of  the  planetary  bodies;  it  is  also  well  worthy  of
attention  on  account  of  the  important  and  exten-
sive observations  which  have  been  made  in  order
to  verify  it.  The  Author  has  put  both  these  sub-
jects together,  commencing  with  the  Figure  of  the
Earth,  and  introducing  Laplace's  Coefficients  when
occasion  required  them;  this  being  perhaps  the  best

VI  PREFACE.

and  simplest  way  of  exhibiting  the  nature  and  use
of  these  coefficients.

The  Author  has  treated  some  parts  of  these  sub-
jects differently  from  the  manner  in  which  they
are  usually  treated,  and  he  hopes  that  by  so  doing
he  has  avoided  some  intricate  reasoning  and  trou-
blesome calculation,  and  made  the  whole  more
accessible  to  students  of  moderate  mathematical  at-
tainments than  it  has  hitherto  been.

In  calculating  the  attractions  of  the  Earth  on
any  particle,  he  has  arrived  at  the  correct  results,
without  considering  diverging  series  as  inadmissible ;
and  this  he  conceives  to  be  important,  because  there
is  evidently  no  good  reason  why  a  diverging  series
should  not  be  as  good  a  symbolical  representative  of  a
quantity  as  a  converging  series ;  or  why  there  should
be  any  occasion  to  enquire  whether  a  series  is  di-
verging or  converging,  as  long  as  we  do  not  want
to  calculate  its  arithmetical  value  or  determine  its
sign.  Instances,  it  is  true,  have  been  brought  for-
wprd  by  Poisson*  in  which  the  use  of  diverging
series  appears  to  lead  to  error;  but  if  the  reason-
ing employed  in  Chapter  in.  of  these  Tracts  be  not
incorrect,  this  error  is  due  to  quite  a  different  cause ;

•  See  Bowditch's  Laplace.  Vol.  H.  p.  167.

PREFACE.  Til

as  will  be  immediately  perceived  on  referring  to  Ar-
ticles 33,  34,  35,  and  37.

The  Author  has  deduced  the  equations  of  motion
of  a  rigid  body  about  its  center  of  gravity  by  a
method  which  he  hopes  will  be  found,  less  objec-
tionable than  that  in  which  the  composition  and  re-
solution of  angular  velocities  are  employed,  and  less
complex  than  that  given  by  Laplace  and  Poisson ;
he  has  also  endeavoured  to  simplify  the  application
of  these  equations  to  the  case  of  the  Earth.

In  the  First  Part  of  these  Tracts  he  has  confined
himself  to  the  most  prominent  and  important  parts  of
each  subject.  In  the  Second  Part,  which  will  shortly
be  published,  he  intends,  among  other-  things,  to  give
some  account  of  the  controversies  which  Laplace's  Co-
efficients have  given  rise  to ;  to  investigate  more  fully
the  nature  and  properties  of  these  functions ;  to  give
instances  of  their  use  in  various  problems ;  for  this
purpose  to  explain  the  mathematical  theory  of  Elec-
tricity ;  to  consider  more  particularly  the  Equations
of  motion  of  a  rigid  body  about  its  center  of  gra-
vity, and  the  conclusions  that  may  be  drawn  from
them ;  to  give  the  theory  of  Jupiter's  Satellites,  and
of  Librations  of  the  Moon ;  'and  to  say  something
on  the  subject  of  Tides.

Vlll  PREFACE.

The  Author  has  not  given  the  investigation  of
the  effect  of  the  Earth's  Oblateness  on  the  motions
of  the  Moon,  but  he  has  endeavoured  to  prove  that
this  effect  does  not  afford  any  additional  evidence
of  the  Earth's  original  Fluidity  beyond  that  which
may  be  obtained  from  the  Figure  of  the  Earth,  and
Law  of  Gravity.

MATHEMATICAL   TRACTS,
PART    I.

CHAPTER    I.

FIGURE    OF    THE    EAUTH.

1.  IT  has  been  well  ascertained,  by  extensive  and
accurate  geodetical  measurements,  that  the  general  figure  of
the  Earth  is  that  of  an  oblate  surface  of  revolution,  de-
scribed about  the  axis  of  diurnal  rotation :  and  this  fact
suggests  the  idea,  that  the  diurnal  rotation  may  be  in  some
way  or  other  the  cause  of  this  peculiar  figure,  especially
if  we  consider  that  the  Sun  and  planets,  which  all  rotate
like  the  Earth,  appear  also  to  have  the  same  sort  of  oblate
form  of  revolution  about  their  axes  of  rotation.

The  most  obvious  and  natural  way  of  accounting  for
the  influence  thus  apparently  exerted  on  the  figures  of
the  planetary  bodies  by  their  rotation,  is  to  suppose  that
they  may  once  have  been  in  a  state  of  fluidity ;  for,  con-
ceive a  fluid  gravitating  mass  to  be  gradually  put  into  a
state  of  rotation  round  a  fixed  axis :  it  is  evident  that  be-
fore the  motion  commenced  it  would,  according  to  a  well-
known  hydrostatical  law,  be  arranged  all  through  in  con-
centric spherical  strata  of  equal  density ;  but  on  the  motion
of  rotation  commencing  a  centrifugal  force  would  arise,
which  would  be  greater  at  greater  distances  from  the  axis,
and  would  therefore  evidently  produce  an  oblateness  in  the
forms  of  the  strata,  leaving  them  still  symmetrical  with
respect  to  the  axis.  Thus  the  hypothesis  of  the  original
fluidity  of  the  bodies  of  our  system,  considered  in  connection
with  their  rotation,  accounts  for  their  oblate  form.
1

2

2.  To  account  for  the  present  solidity  of  the  surface
of  our  own  planet,  we  may  suppose  that  its  temperature  was
originally  so  great  as  to  keep  it  in  a  state  of  fusion,   and
that   this  was  the  cause  of  its  fluidity ;    but  that,  in   the
course  of  ages,  it,  at  least  its  surface,  has  cooled  down  and
hardened  into  its  present  consistence.     This  supposition  is
borne  out  by  geological  facts ;   and  it  is  by  no  means  un-
likely, if  we  consider  that  the  principal  body  of  our  system
is  at  present  most  probably  in  a  state  of  fusion.

3.  This  hypothesis  of  the  Earth's  original  fluidity  re-
ceives much  confirmation  from  observations  on  the  intensity
and  direction  of  the  force  of  gravity ;   for  it  follows  from
the  hydrostatical  law  already  alluded  to,  that  the  Earth,  if
fluid,  ought  to  consist  entirely  of  equidense  strata  of  the
same  sort  of  form  as  the  exterior  surface*,  and  therefore
the  whole  mass  ought  to  be  arranged  symmetrically  with
respect  to  the  axis  of  rotation,  and  nearly  so  with  respect  to
the  centre  of  that  axis.      Hence,  the  force  of  gravity,  which
is  the  resultant  of  the  Earth's  attraction  and  the  centrifugal
force,  ought  to  be  the  same  at  all  places  in  the  same  latitude,
and  nearly  the  same  at  all  places  in  the  same  meridian.

Moreover  it  follows  from  another  hydrostatical  law,  that
the  direction  of  this  force  of  gravity  ought  to  be  every
where  perpendicular  to  the  surface.

Now  all  this  has  been  proved  to  be  the  case  by  nume-
rous observations  with  pendulums,  plumb-lines,  levels,  &c.
(omitting  very  small  variations,  which  may  be  easily  ac-
counted for  in  most  cases).  Hence,  the  hypothesis  of  the
Earth's  original  fluidity  is  confirmed  by  the  observations
which  have  been  made  on  the  force  of  gravity.

4.  But  this  hypothesis  has  been  advanced  almost  to  a
moral  certainty,   by  investigating   precisely   what  effect  it
ought  to  have,  if  true,  on  the  arrangement  of  the  Earth's

*  We  suppose  the  Earth  to  be  heterogeneous,  because  the  pressure  of  the
superincumbent  mass  must  condense  the  central  parts  more  than  the  superh'cial ;
besides,  the  well-known  fact  of  the  mean  density  of  the  whole  Earth  being  greater
than  the  density  of  the  superficial  parts,  proves  that  the  Earth  is  not  homo-
geneous.

mass,  and  by  comparing  the  result  with  observation;  for  it  is
found  that  if  the  hypothesis  be  true,  the  strata  which  com-
pose the  Earth  ought  to  have  not  only  an  oblate  form,  but
one  very  peculiar  kind  of  oblate  form ;  and  it  is  found  that
this  result  admits  of  most  satisfactory  comparison  with  ac-
curate and  varied  observation,  and  actually  coincides  with
it  in  a  most  remarkable  manner;  from  which  we  may  con-
clude, almost  with  certainty,  that  the  hypothesis  is  correct ;
for  it  is  extremely  difficult  to  account  in  any  other  way  for
so  marked  an  agreement  with  observation  of  such  a  very
peculiar  result.

5.  The  object  of  the  following  pages  is  to  give   an
account  of  this  interesting  investigation,  and  to  state  briefly
the  manner  in  which  its  result  may  be  tested  by  observation.
In  the  first  place,  we  shall  determine  the  law  of  arrange-
ment of  the  Earth's  mass,  on  the  hypothesis  of  its  original
fluidity,  by  means  of  Laplace's  powerful  and  beautiful  Ana-
lysis;  and  in  the  next  place,  we  shall  deduce  such   results
as  shall  admit  of  immediate  comparison   with   observation.
The  most  important  of  these  results  are ;   The  expression
for  the  length  of  a  meridian  arc  corresponding  to  a  given
difference   of  latitude,  and.   The  law  of  variation  of  the
force  of  gravity  at  different  points  of  the  Earths  surface.

The  other  results  which  we  shall  deduce  depend  on  certain
assumptions  respecting  the  law  of  density  of  the  Earth,  and
are  therefore  not  so  important.  We  now  proceed,  in  the
first  place,  to  determine  the  law  of  arrangement  of  the
Earth's  mass,  as  follows.

6.  A   heterogeneous  jluid    mass    composed   of   par-
ticles which  attract  each  other  inversely  as  the  square  of
the   distance   rotates    uniformly   in   relative  equilibrium*
round  a  fixed  axis :  to  determine  the  law  of  its  arrange-
ment.

Take  the  axis  of  rotation  as  that  of  #,  and  let  xyz
be  the  co-ordinates  of  any  particle  \$m,  XYZ  the  resolved

*  By  relative  equilibrium  we  mean  that  the  particles  of  the  mass,  though
actually  moving,  are  at  rest  relatively  to  each  other.

attractions  ol'  the  mass  on  \$w,  p  and  p  the  density  and
pressure  at  the  point  (#ysf),  and  o>  the  angular  velocity
of  the  mass.  Then,  by  the  principles  of  Hydrostatics,
we  have

dp  =  p  \Xdx  +  Ydy  +  Zdz  +  w8  (xdx  +  ydy)}  ...  (l)
To  calculate  the  expression  (Xdcc  +  Ydy  +  Zdz),  let  \$m
be   any  attracting  particle,  and  x'y'%  its  co-ordinates  ;   then
we  have

and  similar  expressions  for   Y  and  Z.

Now  assume  (F)   to  denote  the  expression

VV-  *)2  +  (y-  y?+  (*'  -  *)2  '

i.  e.  the  sum  of  each  particle  divided  by  its  distance  from
§m.     Then  it  is  evident  that

y     dV  dV  dV

=  Tx*         ~~dy*         ~dx'

and  Xdfc  +  Ydy  +  Zdz  =  ——  dx  +  —:—dy  -\  --  dx.
dx  dy    '       dss

and  therefore  the  equation   (1)   becomes

The  coefficient  of  p  here  is  a  complete  differential  ;
hence  by  the  principles  of  Hydrostatics,  the  necessary
and  sufficient  conditions  of  equilibrium  are,  that  the  whole
mass  be  arranged  in  strata  of  equal  density,  the  general
equation  to  any  one  of  them  being

C  being  a  constant  different  for  different  strata,  the
exterior  surface  being  one  of  these  strata,  since  it  is  a
free  surface.

7-     Hence  the   equations   from   which   the    problem  is
to  be  solved  are

(A).

=

8.  These  equations  are  unfortunately  very  much  in-
volved in  each  other,  so  much  so  as  to  be  scarcely  manage-
able ;  for   V  must  be  found  by  integration  between  limits
which   depend  on   the  form    of  the  exterior   stratum,  and
therefore  on  the  equation  (A)  ;  and  also  the  law  of  density,
and  therefore  the  form  of  the  internal  strata,  and  therefore
the  equation  (A),  must  be  known  in  order  to  calculate  F.
But   V  is   itself  involved   in   (A),    hence    (A)    cannot    be
made  use  of  in   calculating  V.      It  will  therefore  be  neces-
sary to  devise  some  way  of  eliminating  F,  without  knowing
what  function  it  is.      To  do  this  in  the  general  case  is  be-
yond the  present  powers  of  analysis  ;   but  in  the  particular
case  we  are  concerned  with,  the  fact  of  the  strata  being  all
nearly  spherical,  introduces  considerable  simplification,  and
by  using  the  ingenious  analysis  due  to  Laplace,  we  shall
be  able  to  eliminate  V  with  comparative  ease,  at  least,  ap-
proximately, but  with  quite  sufficient  accuracy.

9.  In  the  first  place,  the  strata  being  nearly  spherical,
we  shall  find  it  convenient  to  make  use  of  polar  instead  of
rectangular  co-ordinates,  and  we  shall  accordingly  transform
our  equations  as  follows  :

Let  r,  0,  (f>,  /,  0',  0',  be  the  co-ordinates  of  \$m  and  \$m'
respectively;  r,  0,  0,  signifying  the  same  as  in  Hymers1
Geometry  of  three  dimensions,  page  (77).  Then  we  have

off  =  r  sin  9  cos  d),

y  =  r  sin  0  sin  <^>,

%  =  r  cos  9,
and  similar  expressions  for  v't  y',  %'.

Hence  the  equation  (A)  becomes
C  =  V  +  —  r-  sin2  9,

2

and  the  equation  (/?)  becomes,  observing  that
\$m  =  p  r'2  sin  9'  dr  d&  dfi

c^  r

Jo     Jo

\Xr'-2rr'|cos#cos#'+sin0sin0'cos(0-^)')}-fr'2

0  and  r,  being  the  limits  of  r',  0  and  TT  of  9',  and  0  and
2?r  of  0',  7*1  being  the  value  of  r  at  the  surface,  and  there-
fore in  general  a  function  of  0'  and  <p'.  It  will  of  course
on  this  account  be  necessary  to  integrate  first  with  respect
to  /,  but  it  is  no  matter  in  what  order  we  perform  the
integrations  relative  to  0'  and  <p',  since  their  limits  are
constants.

If  these  integrations  could  be  performed,  V  would  come
out  a  function  of  r,  0,  and  <^>,  and  unknown  constants  de-
pending on  the  form  of  the  strata  and  the  law  of  density.

10.  We  shall  find  it  convenient  to  put  /*  and  /u?  for
cos  6  and  cos  9'  respectively,  this  will  give

sin  9'd9'  =  -  dp.',

and  the  limits  of  /u.  will  be  —  1  and  1  ;  or  we  may  put  dfi
instead  of  —  d//,  if  at  the  same  time  we  reverse  the  limits
of  fji.  Hence  our  equations  become

(A'),

When  for  brevity  we  have  put

cos  9  cos  ff  +  sin  9  sin  9'  cos  (0  -  <^>')  =  p,
i.  c.     fifj!  +  \/l  -fj?  .  \A  -  //*  • cos  (^>  ~  0')  =  P-

11.  We  shall  now  introduce  into  these  equations  the
condition  that  the  strata  are  nearly  spherical.  If  the  strata
were  actually  spherical,  the  whole  mass  would  be  symmetri-
cal with  respect  to  the  centre,  and  therefore  V  being  the
sum  of  each  particle  divided  by  its  distance  from  \$m  would
depend  simply  on  the  distance  of  \$m  from  the  centre,  and
therefore  be  the  same  at  all  points  of  the  same  stratum.
We  may  hence  conclude,  that  if  the  strata  instead  of  being
actually  spherical  be  only  nearly  so,  V  also,  though  not
actually  the  same,  will  yet  be  nearly  the  same  at  all  points
of  the  same  stratum.  Now  the  value  of  V  for  any  stratum
is  given  by  the  equation  {A')  i.  e.

but  V  (as  we  have  shewn)  ought  to  be  nearly  constant  at
all  points  of  this  stratum,  hence  the  variable  part  of  it,  viz.

—  r2(l  —  /u2)  must  be  always  small  :   therefore  since  r2(l  —  fj?)

is  not  always  small,  wz  must  be  so.  We  shall  accordingly
take  ft>2  as  the  standard  small  quantity  in  our  approxima-
tions, neglecting  its  square  and  higher  powers  in  the  first
approximation.

12.  Now  to8  being  a  small  quantity,  we  may  suppose
the  equation  to  any  nearly  spherical  surface,  and  therefore  to
any  of  the  strata,  to  be  put  in  the  form

where  a  is  the  radius  of  any  sphere  which  nearly  coincides
with  the  stratum  (that  sphere,  suppose,  which  includes  the
same  volume  as  the  stratum*),  and  aa?u  is  the  small  quan-
tity to  be  added  to  a  to  make  it  equal  to  r,  and  therefore  u
in  general  will  be  some  function  of  /u.  and  0.  Moreover,  u
will  be  a  function  of  a  also,  otherwise  the  strata  would  be
all  similar  surfaces,  which  of  course  we  have  no  right  to
assume  them  to  be  ;  a  may  be  considered  as  the  variable

*  We  make  this  supposition,  at  present,"for  the  sake  of  giving  a  definite  idea
of  what  a  is  ;  hereafter  it  will  be  found  an  advantageous  supposition.

8

parameter  of  the  system  of  surfaces  which  the  strata  con-
stitute. We  shall  introduce  the  variable  a  into  our'  equa-
tions instead  of  r,  and  get  every  thing  in  terms  of  a,  /x
and  <£,  instead  of  r,  /u,  and  0;  the  advantages  of  this  change
will  soon  be  perceived.

13.  First,  then,  in  the  equation  (^'),  putting  a  (l+a>2w)
instead,  of  r,  and  neglecting  the  squares  and  higher  powers
of  <*r,  we  find

Next  we  shall  make  a  similar  substitution  in  the  equa-
tion (B')  by  putting  r'  =  a  (l  +  o>2w'),  when  u  denotes
what  u  becomes  when  a,  /*,  and  0  are  exchanged  for  a',  /,
and  <p'  respectively.

Assume  for  brevity,

then  in   the  equation   (B')  we  shall  have

*  =  /oai  p/  (r,  /)         .  da',

where  Oj  is  the  parameter  of  the  exterior  surface.

and  substituting  for  r,f(r,r')=tf{r,(a'+a'u?u)}

dr  9   d(a'u)

and  hence,  since   —  7  =  1  +  to  .  —  -  —  H^  ,    we  have
da  da

9

Hence,  neglecting  the   squares   and   higher  powers  of
ft)2,  the  equation  (B7)  becomes

14.  This  expression  for  V  is  much  more  manageable
than  that  in  the  equation  (B1)  ;  for  in  the  first  place  the
limits  are  all  constant,  and  therefore  we  may  take  them  in
of  depending  on  all  the  variables,  as  of  course  it  did  before,
is  now  a  function  of  one  variable  only,  viz.  a  \  for  each
stratum  being  equidense  throughout,  it  is  evident  that  p  is
the  same  at  every  point  of  the  same  stratum,  and  therefore
varies  only  when  we  pass  from  stratum  to  stratum,  i.  e.  it
varies  with  a  alone.

Hence,  changing  the  order  of  the  integrations,  and
bringing  p  outside  the  integral  signs  relative  to  ^  and  0',
we  have

~    *r   a'u'  d*  <*'       da''

15.  We  have  thus  introduced  into  our  equations  the
condition  of  the  strata  being  nearly  spherical.

We  shall  find  it  convenient  to  make  a  farther  substitu-
tion in  this  equation,  viz.  by  putting  r—a(l  +  <o2w),  which,
neglecting  the  squares  and  higher  powers  of  <o2,  gives

r  P  {r  V-!  /(a'  a/>  dfi'  w

—  ^  f    /(a,  a')  dp.'  d<b'
aaJ<\     J  -\"

'an  (a'  ir  f  f(a*  a'^u>  dfjL  d<p}  \  da''

10

16.  The  next  thing  we  shall  do  is  to  perform  the  in-
tegrations relative  to  /*'  and  0' ;  to  do  this  we  shall  expand
the  quantity^y^aa'),  which,  since  it  represents

a'z
\/a/2-2aa'p+a2'

a         a

may  be  expanded  in  a  series  of  powers  either  of   —  or  — .

a          a

The  coefficients  will  evidently  be  the  same  whether  we  ex-
pand it  in  powers  of  —.  or  of  — ,  they  will  in  fact  be  the
a  a

coefficients   of  the  powers  of  h  in    the    expansion   of  the
quantity

1

We  shall  assume  Qy,  Qn   Q2,  &c.  to  denote  these  co-
efficients, i.  e.  we  shall  assume

Q0  will  evidently  be  unity.     The  rest  of  these  coefficients
will  be  rational  and  integral  functions  of  p,  i.e.  of

fifi   +  V   1   —  yU2   .   V   1   —  //2   .    COS  (0  —  0').

It  is  evident  that  they  all  become  unity  when  p  becomes
unity ;   for  then

becomes   ,    or    1  +  h  +  h2  +  &c.

We  shall  have  no  occasion  however  to  determine  their
forms.  They  (and  other  functions  of  the  same  character)
are  the  celebrated  coefficients  of  Laplace ;  they  possess  very
remarkable  properties,  which  we  shall  now  digress  to  inves-
tigate, as  they  wonderfully  facilitate  the  integrations  we
have  to  perform,  and  enable  us  to  eliminate  V  from  the
equations  (A")  and  (5"'),  with  great  facility,  without  know-
ing its  form.

CHAPTER    II.

LAPLACE    S      COEFFICIENTS.

17.  IN  order  to  investigate  the  properties  of  the
functions  Q,,  Qn  Q2,  SEC.  introduced  in  the  last  chapter,  we
shall  recur  to  the  expression  from  which  they  were  origin-
ally derived,  viz.

1

We  meet  with  this  expression  constantly  in  physical
problems,  especially  those  in  which  attractions  are  con-
cerned, and  it  is  therefore  worthy  of  particular  consi-
deration.

Assuming  R  to  denote  this  expression,  we  have

R

'-  x?  +  (yf-  yf  +  (x'-)*'

and    differentiating  this   equation    twice  relatively  to  xyx
respectively,

dR  __  (x-.v)
dx  ~  K*'-*)2+(y'-y)2  +  (*'-*)2}*
-JP  (*'-*)

«       .JP'V-.)-*

dx*  dx

and  similarly

fp  Tt
—

12

\

hence  evidently

d-JR      d*R      d*R  _          1

*    =0 (1).

18.  We  shall  express  this  differential  equation  in
terms  of  the  polar  co-ordinates  r0(p  instead  of  xyx.  To
facilitate  the  transformation  we  shall  assume  an  auxiliary
quantity  s,  such  that

s  =  r  sin  0 ;

and  therefore  since  x  =  r  sin  0  cos  0,  and  y  =  r  sin  6  sin  0,
we  shall  have

X  =  S  COS  0,

y  =  s  sin  0.

Then    considering  s   and   0  as   independent    variables
instead  of  x  and  y,  we  have
dR  _  dR   dx      dR   dy
ds        dx    ds       dy    ds

dR  dR  .

=  — —  cos  0  +  — —  sm0 (2).

dx  dy

dzR      d*R  dzR  d?R

and  — —  =  — —  cos^0+2   • — —  cos 0 sin  0+  — —  sm20...(3).
ds*       dx2  dxdy  dy2

.  dR          dR  dR

and  -**,=  -  — *  sin  d>  H s  cosd> (4).

(fm         dx  dy

d*R      d*R       .  d*R  d?R

-r  =  -T-r*f  Sin  0  —  2  — —  .    S^COS  0  Sin  0  H S2  COSZ(b

d0         dx2  dxdy  dy2

dR    '  dR

Equations  (3)  and   (5)  give

d?R      l    d*R      d?R      tfR      1  (dR  dR

~TT  +  1   T3T  =  T~T  +  ~T~9  ---      —  cos  0  +  —  —  sin  0
ds*       s2   d02       dx2       dy2       s  \dx  dy

d*R      d?R      1  dR

by  equation  (2).

13

Now   we  have

x  =  r  cos  9
s  —  r  sin  0*,

and  these  equations  connect  %  s  r  9  in  exactly  the  same  way
in  which  xys(f>  are  connected  by  the  equations

X  =  S  COS  0

y  =  s  sin  0.
Hence  we  may  prove  exactly  as  before,  that

dR       l    (PR  _  d'R      d*R  _  i   dR
d^Jr^i~d¥=  "d**"  +  rf*2   ~  r~dr'"
adding  this  to  the  equation  (6),  we  have  by  (l),
I.  tfR      rffl       l^  d~R         l  dR      l   dB_
?  d0*  +~d7+  r2   drF"     «    ds       r~d7"
Now  by   (2)  and  (4),

dR  dR  cos0      dR

—  sin  0  +  —     —2-  =  —  -  ,
ds  d(f)       s  dy

and  hence  observing  as   before,  that   x  s  r  9  are  connected
together  in  exactly  the  same  way  as  x  y  s  0,  we  have
dR    .  dR  cos  9      dR

hence,  substituting  this  value  of  —  —  in  equation  (7),  and

ds

putting  r  sin  9  for  s,  and  multiplying  by  r2,  we  have
d*Jl      cos  9   d_R_         I      d^R.       ,d*R          dR_  _
J¥  +  sin  9    ~d9  +  shT^  dtf  +  r  ~d7  +  'T  ~dr  ~  °'
l       d  dR\          l      d*R       d   f     dR

which  is  the  equation  (l)  expressed  in  polar  co-ordinates.

*  The  author  finds  that  he  has  been  anticipated  in  making  this  use  of  the
auxiliary  quantity  s,  by  the  Cambridge  Mathematical  Journal.

14

19.      Now  K  expressed  in  polar  co-ordinates  becomes
1

Vr    -

which  =  —
r

/  r       r*

V    1    -  %P   -  +  -72

Qn       n  +   &C.

(See  Art.  16).  Hence  substituting  this  value  of  R  in  the
equation  just  obtained,  and  putting  the  coefficient  of  r"
equal  to  zero,  since  r  is  indeterminate,  we  find  imme-
diately

sin  0  dO  \          d9  ]       sin*  0  d(f>

which  is  a  partial  differential  equation  of  the  second  order,
connecting  Qn  with  /a.  and  0  ;  of  course,  being  such,  it
admits  of  an  infinite  number  of  solutions  besides  Qn.
We  shall  have  no  occasion  to  solve  it,  but  "we  shall
find  it  of  use  in  investigating  the  properties  of  Qn.  All
solutions  of  it  which  are  rational  and  integral  functions
of  cos  0,  sin  0,  cos  <f),  sin  <p  (i.  e.  of  /x,  \/l-fj?9  cos  (f),
sin  0),  are  called  Laplace's  coefficients  of  the  nth  order,
having  been  first  brought  into  notice  by  Laplace  in  his
Mecanique  Celeste,  Liv.  in.:  the  equation  itself  may
be  called  Laplace's  equation.  Why  we  restrict  Laplace's
coefficients  to  be  rational  and  integral  functions  of  /u>
'X/l  -  M8>  cos  0  a°d  sin  (f>,  will  appear  presently.

20.     We  may  remark  here  that  in  consequence  of  the
linearity  of  Laplace's  equation,  the   sum  of  any  number  of

15

Laplace's  coefficients  of  the  wth  order  is  also  a  Laplace's  co-
efficient of  the  wth  order.

Also  any  constant  quantity  is  a  Laplace's  coefficient  of
the  order  0,  for  if  F0  be  a  Laplace's  coefficient  of  the  order
0,  we  have

d

which  equation  is  evidently  satisfied  by
F0  =  any  constant,

and  hence  any  constant  is  a  Laplace's  coefficient  of  the
order  0.

It  may  easily  be  seen  by  trial  that

a/i,  and  a  \/l  -  /u2  cos  (0  +  /3)

are  Laplace's  coefficients  of  the  order  1,  a  and  /3  being
constants  ;  and

a  (1  -  M2)>  a  jtA  \A  ~  M2  cos  (0  +  /3),   a  (l  -  fj?)  cos  (2  0  +  /3),

are  Laplace's  coefficient  of  the  order  2,  and  so  on.  We
shall  not  have  any  occasion  at  present  to  determine  the
general  expression  for  a  Laplace's  coefficient  of  the  wth  order,
but,  to  give  clear  ideas,  we  shall  just  state  that  it  may  be
put  in  the  form

A0  Mn+4}  (I  -fjfyM^costy  +  aj  -I-  4,(1  -ffiMn_  2cos(2<p  +  a2)  ,
&c  .............  +A(l-

When  AQ  Al  SEC.  ...  a^  a2  ...  &c.  are  any  constants,  Mn  Mn_^
&c.  contain  rational  and  integral  functions  of  /u,  of  the  di-
mensions w,  n  —  l,  n  -*2,  &c.  respectively*.

21.  The  first  property  we  shall  prove  of  Laplace's
coefficients  is  this  :  If  Ym  and  Zn  be  any  two  Laplace's
coefficients  of  the  mth  and  wth  orders  respectively,  then

i      I     Ym  Zndju.  d(f)  =  0,  except   when   m  =  n.

Jo      J  -i

*  We  shall  recur  to  this  subject  in  Part  n.  of  these  Tracts.

16
For  since  ZM  satisfies  Laplace's  equation,  we  have

n  (n  +  1)      *"       Ym  Zn

Now  integrating  by  parts,   and  observing  that  1  -  M2  =  0,
at  each  limit  we  have

and  similarly,

f"v  fz-j          r~  dr.dz
J,    Y"^d*  =  -  J.    Tp-jf**.

1  /^

*  observing    that    Ym  -•--*   is  the   same  at   each  limit,   be-
ct(p

cause  Ym  and  Zn  are  functions  of  sin  0  and  cos  0,  and  not
of  0  simply  ;   hence  substituting  in  (l)

n.(n  +  l)  r*r

J0     J-\

r^  ri  (/       2,dr»

-I.  L(i-^^'

In  exactly  the  same  way  we  may  shew  that

f'

-I.

*  This  is  the  reason  why  we  have  assumed  Laplace's  coefficients  to  be
functions  of  sin  <f>  and  cos  <7>,  and  not  of  <j>  simply.

17

Hence  subtracting
\n(n  +  1)  -m(m+  1)  }        *         Ym  Zndnd<t>=  0  .

Now  the  factor  n(n  +  l)  —  m(m  +  1)  does  not  =  0,
except  when  m  =  n  ;  hence  the  other  factor  must  be  zero,
hence

r*  r  Ymznd/uid(f>=o,

J0      J  -}
except  when  m  =  n.

22.     Since  Q0  =  l  (see  Art.  16),   we  have

f

J  —

=  0,  when  n  is  greater  than  0,

=  r^  f

'0
=  47T.

It  need  scarcely  be  remarked  that  Q0,  Q15  Q2,  &c.  possess
exactly  the  same  properties  with  respect  to  p  and  0'  that
they  do  with  respect  to  /j.  and  (p.

23.  We  shall  now  have  occasion  to  introduce  a
remarkable  discontinuous  function,  but  before  we  do  so
we  shall  give  a  simple  example  of  functions  of  this
description,  in  order  to  render  our  reasoning  more  satis-
factory to  those  who  have  not  been  accustomed  to  them.

We  may  easily  prove,  by  the  aid  of  the  exponential
value  of  the  cosine  of  an  arc,  that

*(.-f)

cos  a  +  cos  (a  +  /3)  +  cos  (a  +  2/3)  ad  inf. ...  = — ;

suppose  here  that  a  =  — ,  then  we  find

0
cos  a  +  cos  3  a  +  cos  5  a  +  ...  =  -

2  sin  a

18

hence  this  series  is  always  zero,  except  when  a=  any  mul-
tiple of  TT,  in  which  case  sin  a  becomes  zero,  and  therefore

the  series  becomes  - ;   thus  though  each  term  of  this  series

varies  continuously  with  a,  the  series  itself  varies  discon-
tinuously,  being  constantly  zero,  except  when  a  passes
through  any  of  the  values  0,  ±  TT,  ±  2  ?r,  &c..  when  the  series

suddenly  becomes  -;   i.e.,  some  unknown  or  indeterminate

quantity.  To  explain  the  nature  of  this  series  more  clearly,
we  observe  that

sin  —  |  cos  a  4-  cos  (a  +  /5)  +  SEC.  }  =  -  ^  sin  f  a ) ,

whatever   be   the    values    of    a    and    /3 ;     suppose    a  =  —  ;

then  sin  I  a 1   becomes  zero,    whatever  be  the  value  of

a ;   hence

sin  a  {cos  a  +  cos  3a  +  cos  5a  +  &c. }  =  0,
for  all  values  of  a.

Now  as  long  as  a  is  not  a  multiple  of  TT,  sin  a  will  not  be
zero,  and  therefore

cos  a  +  cos  3  a  +  ...  &c.

must  be  zero ;  but  if  a  be  any  multiple  of  TT,  then  sin  a  will
be  zero,  and  our  equation  will  be  satisfied  quite  independent-
ly of  its  other  factor,  and  hence  will  give  us  no  information
as  to  the  value  of  that  factor;  hence  when  a  is  any  multiple
of  T,  cos  a  +  cos  3 a  +  &c.  is  some  unknown  or  indeterminate
quantity.

It  is  important  to  remark  that  the  change  in  the  value  of
cos  a  +  cos  3a  +  &c.,  when  a  becomes  a  multiple  of  -a-,  is
perfectly  sudden,  for  since  the  second  member  of  the  equa-
tion is  always  absolutely  zero,  it  is  evident  that  as  long  as
sin  a  is  not  actually  zero,  though  it  differs  from  it  by  ever  so

19

small  a  quantity,  cos  a  +  cos  3  a  +  &c  ----  must  be  so;  for
this  reason  cos  a  +  cos  3  a  +  Sic.  is  called  a  discontinuous
function.

24.  We  shall  now  bring  forward  the  remarkable  dis-
continuous function  we  alluded  to:  it  is  the  following  series,
viz.

Qo+  3Q/+  &c-  +  (2w  +  1)  Q«+  &c.  ad  inf.
this  series  is  of  exactly  the   same  nature  as  that  we  have
just  considered,  being  a  function  of  the  variables  /x  and  <£»,
which  is  always  zero,  except  for  certain  particular  values  of
these  variables.

To  shew  this,  we  have

Q  +  Q,A  +  ...  Qnh»  +  &c.  =     ,  ~  ,

VI  -  2ph  +  h?

and  differentiating  this  relatively  to  /*and  multiplying  by  2  h,

«,+  »(t*...(g.  +  .)  «.*•+  &c.  -  (1--^¥

now  here  put  h  —  1  and  we  find

0

O,  +  3  Q  +  &c.  =  -  —     —  =  ,
(2  -  2p)f

hence  QQ+  3Q7  +  &c.  =  0  for  all  values  of  />,  except  p

0

when  it  becomes  -  :  now
0

P=fJLfl'  +  \          -/(Z2      ^l    -  /li'2   COS  (         -         '),

but  cos  (0  -  <£')  is  not  greater  than  1  ;

20

hence,

1  -/*/*'
or,  squaring  and  reducing,

(fj.  —  p)*  is   not    >  0,

which  cannot  be  unless  /u.  =  //,  and  this  will  give
cos  (<p  -  <£')  =  1  ;

and  therefore  0  -  0'  is  zero,  or  some  multiple  of  STT;  hence
the  series  Q0  +  3  Qx  +  &c.  is  a  discontinuous  function,  being
always  zero,  except  when  M  =  M'  and  <f>  ~  <p'  =  2wrr,  (m

being  any  integer,)  in  which  case  it  becomes  -.      It  is  evi-

dent, as  in  the  former  case,  that  this  series  is  perfectly
discontinuous,  being  zero  for  all  values  of  p  that  differ  even
in  the  least  degree  from  unity,  and  then  when  p  =  1,  sud-

denly assuming  the  form  -  .

25.  Now,  wherever  we  have  occasion  to  use  the  series
Q0  +  3Ql  +  Sec.,  it  will  occur  under  integral  signs  relative  to
fi  and  0,  and  the  limits  of  <f)  will  be  0  and  STT;  hence,  by
what  we  have  proved  in  the  note*  respecting  the  limits  of

*  If  x  be  any  arbitrary  quantity  occurring  in  any  investigation,  its  differen-
tial dx  may  be  defined  to  be  any  small  increment  of  x,  made  use  of  with  the
understanding  that  it  is  to  be  put  equal  to  zero  at  the  end  of  the  investigation  ;
and  if  f(x)  be  any  function  of  at,  its  differential  df(x)  may  be  defined  to
be  the  corresponding  increment  of  /(#),  that  is,

The  symbol  j  written  before  a  differential,  is  generally    taken  to  denote  the
quantity  from  which  the  differential  is  derived,  that  is,

/«*/(*)  =/(*),

and  the  notation  /**<*/(*)  is  taken  to  denote  the  difference  f(x3)-f(xl):
but  this  notation  has  a  much  more  important  signification  :  for  in  the  equation

put  for  x  successively  the  values  .r,,  Xi  +  dn,   z1  +  2dn,  &c.,   #,  +  (»-  \)dx,
and  add  the  results,  and  we  find

!  +  dx)  +  d/(-r,  +  2dx)  &c.  +  df{Xl  +  (n  -  1)  dx\

21

integrals,  0  will  receive  all  the  values  between  0  and  27r,
inclusive  of  the  inferior  limit,  and  exclusive  of  the  superior,
and  will  therefore  never  actually  be  equal  to,  or  exceed  2?r.
Also  in  all  cases  we  shall  be  concerned  with  the  same
may  be  supposed  true  of  <p'  ;  for  in  all  our  investigations,
wherever  0'  recurs,  the  value  <f)'  =  2  IT,  or  any  greater  quan-
tity, will  be  only  a  repetition  of  <f>'  =  0,  or  some  value
between  0  and  2  TT  ;  hence  we  may  consider  that  <f>  ~  <f>
never  actually  equals  or  exceeds  2-rr;  and  hence  it  will  be
only  for  one  value  of  0  ~  0',  viz.  0,  that  p  will  become
unity  ;  hence,  by  what  we  have  proved,  the  series
Q0  +  3Q,  +  &c.

in  all  cases  we  shall  be  concerned  with,  will  be  absolutely
zero  for  all  values  of  fj.  and  0,  except  the  single  values
[A.  =  p  and  0  =  0'.  We  now  proceed  to  prove  some  re-
markable properties  of  this  series,  which  result  from  its
discontinuous  nature.

26.      From  Art.  22  it  appears  immediately  that

=    4<7T.

Now,  here  the  quantity  under  the  integral  signs  is,
as  we  have  proved,  always  zero,  except  when  p.  =  /m'  and
0  =  0';  it  is  therefore  no  matter  what  the  limits  of  the
integration  be,  provided  they  include  between  them  the

or  supposing  >rj  +  ndx  =  ,r2,
df(xt)  +  df(xl  +  dx)+&c.  till  we  come  to  df(xa-dx)=f(xs)-f(xl)

=  £«<*/(*).

Hence  it  appears  that  if*8  df(x)  denotes  the  sum  of  a  series  of  values  of  df(x),
got  by  giving  x  all  its  values  between  the  limits  xt  and  xz  inclusive  of  the
former  limit  and  exclusive  of  the  latter;  that  is  to  say,  all  the  values  of  x
which  form  an  arithmetic  series  whose  common  difference  is  dxy  commencing
with  xl  and  ending  with  xz-  dx.  The  remark  respecting  the  limits  is  im-
portant whenever  discontinuous  functions  are  concerned,  as  in  our  present  in-
vestigation ;  and  we  must  remember  that  though  the  last  value  of  x  approaches
indefinitely  near  to  x2,  it  never  actually  becomes  equal  to  it.

alues  (fji  =  /u')  and  (0  =  0'),  respectively  ;    hence,  if  /*,,  ^2,
i»  02*5  be  any  limits  which  do  this,   we  have

*  (Qn  +  3Q!  +  &c.  ...)dnd<b  =  ±Tr.

27-  In  the  same  manner,  if  F  (n<p)  be  any  function
of  /u  and  0,  which  is  always  finite  between  the  limits  -  1
and  1,  0  and  Zir,  F(yu</>)(Q0  +  3Q1  +  &c.)  will  be  always
zero,  except  when  JM  =  //  and  0  =  0',  and  we  shall  have,
as  before,

/^2  rM

•^l    »^l

28.  Now  let  F(n"(f>")  be  the  greatest  value  of  F(^0),
between  the  limits  /ui,  yua»  0i>  02  '•>  ano<  let  ^'(/*//0//)  be  the
least ;  then  it  is  evident  from  the  nature  of  an  integral,
considered  as  a  sum,  that

/-</>*    rn*  F  ^/0/j  ^QQ  +  3  Q!  +  &c)  d^d0
is  not  greater  than

.F(/u"0")  y       T  2  (Q0  +  3Q,  +  ...)  dyurf0,
and   not  less  than

i.  e.  (by  Art.  22),

not  greater  than   4  ?r  F  (^"0"),

and  not  less  than  4  TT  T'1  (^^0^)  ;

and  this  is  true,   no  matter  how  close  together  the  limits

*  Of  course  these  limits  are  supposed  to  be  included  between  -  1  and  +  1,
0  and  2ir.

/"u  Us*  <£n  ^a?  be  taken,  provided  /UL'  and  (j)'  be  included  be-
tween them.  Now  /*",  0",  and  /u//5  0//s  are  also  always
included  between  these  limits  ;  hence,  since  /&'<£',  M">  </>"»
yu//s  0/x,  are  respectively  always  included  between  limits
which  we  may  take  as  close  together  as  we  please,  it  is
evident  that  we  may  suppose  //',  0",  and  //„»  0y/»  to  differ
from  yu'0'  respectively  by  as  small  quantities  as  we  please ;
and  therefore,  since  F  (/u.  <p)  is  always  finite,  we  may  in
the  above  inequalities  suppose  F(/JL'>  <^>")  and  F  (p.^  <£„)
as  nearly  equal  to  F(fj!<p')  as  we  please,  which  evidently
cannot  be,  unless

FT

jfr  jfr

29.      We    shall    give    another    demonstration    of    this
remarkable  result.

Assume,  as  we  evidently  may,
(Q0+3Q1+  ...

Then,  as  before,

r*  T

fc/Q  •/   —  1

multiply   this  equation  by  d//  d^)',  and  integrate  between
the  same  limits,  and   we  have*

It  is  necessary  to  take  the  same  limits,  otherwise  in  the  integral

/i  and  ^>,  the  values  of  the  variables  for  which  Q0+3Qi  +  ...  becomes  «,  will
not  be  always  included  between  the  limits,  and  therefore  Art.  2fi  will  not  apply
to  it,  and  our  proof  will  be  incorrect.

24
Hence,  by  Art.  26',

differentiating  this  equation  relatively    to   <p.,  and   yu2  suc-
cessively*,  we  have

4-TT  F  (fJL2(f)o)    =,/*(Al2</>2);

hence,  since  ^2  and  02  are  arbitrary  ,y=  47T-F;  and  therefore
(Q0  +  3Q,+  ...)  dfjidQ

30.     We  may  hence  find  the  value  of
TV'  YnQndndd).

Jn       J  -i

Yn  being  any  Laplace's  coefficient  ;  for  Yn  being  a  rational
and  integral  function  of  /m,  \/l-to>2,  cos  ^>,  sin  0-f-  will  be
always  finite  ;  hence  we  may  put  Yn  for  F  (/a.  (pi)  in  Art.  28,
and  we  find  immediately  by  Art.  21,

PV  YnQn

J0      J-i

*  To  shew  how  to  differentiate  a  definite  integral  with  respect  to  its  limits,
let  f(x)  +  C  denote  the  indefinite  integral  off'(x),  then

differentiating  this  relative  to  <r2,  we  have

and  differentiating  relative  to

and  in  a  similar  manner  we  may  differentiate  integrals  relative  to  two  or  more
variables.

t  This  is  the  reason  why  we  have  restricted   Laplace's  coefficients  to  be
rational  and  integral.

25

where  Y '  denotes  what  Y  becomes,  when  /tx.'  and  <f>  are  put
for  (ix  and  (f>  in  it.

This  is  a  very  important  result ;  in  fact,  this,  and  that
in  Art.  21,  are  the  properties  which  render  Laplace1  s
coefficients  so  very  useful  in  integrations  such  as  we  have
to  perform  in  Art.  15.

31.  The  equation  deduced  in  Art.  28,  interchanging
fjL  and  (f)'  for  /tx  and  <£,  shews  that  if  F(fj.(j))  be  any  func-
tion of  fjL  and  (p,  which  is  always  finite,  it  may  be  expanded
in  a  series  of  Laplace's  coefficients ;  for  by  this  equation
F  (n<p)  =  a  series  whose  general  term  is

2n  +  1    rt  tr  /M          ,    ,  ,

—     /  I        F  (/JL  <p  )  Qn  d  fJL    U  (t)  .

4>ir      J o     »'— i

Now  this  quantity  evidently  satisfies  any  linear  differ-
ential equation  relative  to  fj.  and  0  that  Qn  satisfies ;  there-
fore it  satisfies  Laplace's  equation  of  the  wth  order  ;  moreover
it  is  a  rational  and  integral  function  of  /x,  \/l  — ti2,  cos  <£,  sin<^>,
for  Qn  is  so,  and

2W+    1        /-8W     /M      r

will  evidently  differ  from  QB,  considered  as  a  function  of
^,  \/l-/x2,  cos  0,  sin  0,  only  in  having  different  coefficients
to  the  powers  of  these  quantities  ;  that  is  to  say,  if  A'
be  any  coefficient  in  Qn,  then  the  corresponding  coefficient  in

')  Q. dp' dip

27T

will  be

hence  the  several  terms  of  the  series  to  which  F'(/u'(p')  is
equivalent  are  rational  and  integral  functions  of  /u,  \/l-/r,
cos  0,  and  sin  0,  which  satisfy  Laplace's  equation,  and

4

are    therefore,   according    to    our    definition,    Laplace's  co-
efficients.

32.     No   function  can  be  expanded  in   more  than   one
series  of  Laplace's  coefficients.

For,  if  possible,   let

Y0  +  Yl  +  ¥„  +  ...  +  &c.  and  Z0  +  Zl  +  Z2  +  &c.

be  two  different  series  of  Laplace's  coefficients  equivalent  to
the  same  function,  then,  since  this  is  the  case,  we  have

Y0  -  Z0  +  Y,  -  Zl  +  Y,  -  Z2  +  &c.  =  0;

multiplying  this  equation  by  Qnd/i.d<p,  and  integrating  be-
tween the  limits  -  1,  1  ;    0,  27r,  we  find  (by  Art.  22  and  30),

(Y'n  -Z'a)  =0,

"2n  +  1

hence  Yn  =  Zn ;  and  therefore  the  series  are  the  same,  and
the  function  can  be  expanded  only  in  one  series  of  Laplace's
coefficients.

33.     The  conclusion  we  have  just  arrived  at  seems  to
be  at  variance  with  the  fact  that  the  quantity

may  be  expanded  in   two  distinct   series  of  Laplace's  co-
efficients, viz.

but  this  is  only  an  apparent  discrepancy,  for

the  function  to  be  developed,  admits  of  two  values,  one
positive  and  the  other  negative,  on  account  of  the  ambiguity
of  the  sign  of  the  square  root  ;  and  therefore  ought  to  admit
of  two  developments.  That  the  above  are  the  two  develop-
ments corresponding  to  the  two  values  of  the  square  root,
will  follow  from  the  following  proposition  :  viz.

27
34.     To  determine  the  sign  of  the  series

This  series  being  the  development  of  the  expression
1

it  is  evident  that  it  can  never  become  0  as  long  as  h  is  not
infinite,  but  it  will  become  infinite  when

1  -  2ph  +  h2  =  0  ;

2
that  is,  when  p

.

Now  1  +  h2  is  always  greater  than  2A,  except  when  h  «  1,
in  which  case  1  +  A2  =  2A;  hence,  since  p,  being  a  cosine,
can  never  exceed  unity,  this  equation  can  only  be  satisfied
by  h  =  1  ,  and  therefore  p  =  1  ;  hence

cannot  become  infinity  unless  h  be  unity  ;  hence,  if  h  be  not
infinity  or  unity,  this  series  can  never  become  zero  nor  in-
finity, and  therefore  can  never  change  its  sign  :  supposing
then  that  h  is  not  infinity  or  unity,

will  always  have  the  same  sign  whatever  be  the  value  of  p.
Hence,  supposing  p  =  1,  and  therefore  Q0  =  1,  Q1  =  1,  &c.
(Art.  16.)  it  is  evident  that

Qo  +  Qih  +  &c.  has  the  same  sign  as  1  +  h  +  h?  ...

1

i.  e.  as  -  -;
1  -h

hence,  if  h  be  less  than  unity,  this  series  is  positive,  and  if
h  be  greater  than  unity,  it  is  negative.

In  the  same  way  it  may  be  proved  that

tilc  samc  8'n  as

\  7         g  -r*  7  -  '

rtf  ft  ft    -~    1

and  is  therefore  always  negative  when  h  is  less  than  unity,
and  positive  when  h  is  greater  than  unity.

Hence,  when  h  is  less  than  unity,

i
QO  +  Q\h  +  Qzh2  +  is  the  positive  value  of

V  1  -  2ph  +  tf

and  -  \  Qo  +  Qi  7  +  . . .  \  the  negative  value ;
h  (  h  }

and  when  h  is  greater  than  unity,  the  reverse  is  the  case.

35.     To  represent,  then,  the  true  general  development

of  —j-  ,  let  A;  be  a  discontinuous  function  of  h,

Vl  —  %ph  +  h*

such  that  A:  =  1,  when  h  is  less  than  unity;

and  k  =  0,  when  h  is  greater  than  unity.

Then,  whatever  h  be,  the  positive  value  of  — ^

V  ]

will  evidently  be

and  the  negative  value  will  be
(1  -k}  {Q0+  Q,h  +  Q2A2+  ...}  +  k  JQ0^  +  Q  ^+  ...1.

Thus,  in  reality,       ,  ,   if  we   restrict  our-

Vl-2ph  +  hz

selves  to  only  one  of  its  values,  can  be  developed  in  only
one  series  of  Laplace's  coefficients.  The  conclusion  we  have
just  arrived  at  respecting  the  true  development  of  this
quantity,  will  be  highly  important  in  our  future  investi-
gations.

36.     We  shall  conclude  this  chapter  with  the  following
important  property  of  Laplace's  coefficients,  viz.

29

If         Y0+Yl  +  F2  +  &c.  =  0  (l),

be  any  equation  arranged  in  a  series  of  Laplace's  coeffi-
cients, IJL  and  0  being  indeterminate,  then  we  must  have

F0  =0,    Y,=  0,    F2  =  0,  &c.

for,  multiplying  (l)  by  Q,,dfj.d(p,  and  integrating  between
the  limits  -  1,  1,  and  0,  27r,  we  have,  (by  Arts.  22  and  30,)

2tt  +  1
and  therefore   Yn  =  0.

We  have  now  considered  these  remarkable  functions  at
sufficient  length  for  our  present  purpose ;  we  shall  recur  to
this  subject  in  Part  u.  of  these  Tracts.

CHAPTER    III.

FIGURE     OF     THE     EARTH.

37.  WE  are  now  prepared  to  return  to  the  equation
(5'"),  see  Art.  15  ;  the  properties  we  have  proved  La-
place's coefficients  to  possess,  will  enable  us  to  perform  the
integrations  relative  to  //  and  <£'  with  great  facility.

In  Art.  16  we  stated  that  we  should  expand  the  quan-
tity f  (a,  a'),  or  /  ,  in  either  of  the  series

Now  there  is  an  ambiguity  in  this  quantity,  since  on  account
of  the  square  root,  it  admits  of  two  values,  one  positive
and  the  other  negative  ;  but,  on  referring  to  Art.  6,  it  is
evident  that  the  square  root  is  always  supposed  to  have  its
positive  value;  for  the  distance  between  \$m  and  \$TW',  which
is  expressed  by  this  square  root,  is  evidently  taken  to  be  the
absolute  or  numerical  distance,  without  reference  to  sign  ;
hence,  by  what  has  been  proved  in  Art.  35,  neither  of  the
above  series  will  give  the  true  general  development  of
y(«,  a'),  but  we  must  put

/(a,  a')  =  *.          Q0  +  Q       +  Q2       +  &c.
+  (1  -  fc)a'  JQ0  +  Q,^  +  Q2^+  &c.},
where  k  is  a  discontinuous  function  of  —  ,  which  is  always

unity  while  a  is  less  than  a,  and  zero  when  a  is  greater
than  a.  For  brevity  we  shall  take  An  to  denote  the  co-
efficient of  Qn  in  this  series  ;  i.  e.

and  then,   we  shall  have

38.  What  we  have  proved  of  Laplace's  coefficients
naturally  suggests  the  advantage  of  expanding  the  quan-
tity u  in  a  series  of  Laplace's  coefficients  ;  this  may  be
done,  since,  on  account  of  the  assumption  we  have  made
respecting  the  nearly  spherical  form  of  the  strata,  u  can
never  become  infinite  ;  we  shall  assume,  therefore,

u  =  u'0  +  u\  +  u'2  +  &c.

tt'o,  M'J,   &c.    being  Laplace's  coefficients  of  the  orders  0,
1,  2,  SEC.   and  functions  of  a',  /u',  and  (£'.

We  shall  denote  by  uot  ult  u2,  &c.  what  w'Q,  u  ^
u'2,  &c.  ...  become  when  a,  yu,  and  (f>  are  substituted  for
o',  /n',  0',  and  therefore  we  shall  have

u  =  u0  +  ul  +  u2  +  &c.

We  shall  also  have  occasion  to  make  use  of  the  values
of  w'0,  7/n  w'j>,  Sec.  when  /u.  and  (f)  alone  are  put  for  /u.'
and  0',  a'  remaining  unaltered,  These  values  we  shall
denote  by  'w0,  fuiy  'w2»  &c-

39.

find

and

J9.      Hence  in   the  equation   (5'"),    see  Art.   15,    we
immediately  by  Arts.  22  and   30,

f

32

and  hence,  putting  u0  +  w,  +  w2,  &c.  for  u,

y  _  47r  /  -•  y  i   ^    .      2  _  /..    ...    .   ..    .   o—  \  a^o

=  4?r  /    '  p  A0da
•/«

+  a  series  whose  general  term  is

4-7T  <y*  f  '  p'  {  aun  —  —  +  -   —  -.  (a  An'un)\da  .
Jo          \       n   da       2n  +  1   do'  v  J

40.      If  we  put  for  A0  and  .4n  their  values,  viz.

a'2  an+2  a"

k  —  +  (l-k)a',  and  k  —  +  (1  -  A?)    ^  ,

(see  Art.  37.)  ;  the  factor  of  47rfc>2  in  this  general  term  be-
comes

Now  /c  is  always  unity  while  «'  varies  between  the
limits  0  and  a,  and  zero  while  a  varies  between  the  limits
a  and  al  (a  being  of  course  never  greater  than  a^  ;  hence
that  part  of  the  quantity  under  the  integral  sign  which  is
multiplied  by  k  will  not  exist  except  between  the  limits
0  and  a,  and  that  part  multiplied  by  1  -  k  will  not  exist
except  between  the  limits  a  and  al  ;  hence  this  integral  may
evidently  be  put  in  the  form

-UJLf*  p'a'2da'+-r  ~  r  (a  p  —  (XXn+s)  da

a  Jo    ^  (2n  +  1   an+l  J0    r  da'  y

t  ,  d  f  'u.  \
p  —,  \~-.  }
^  da  \an-2J

33

We  shall,  for  brevity,  denote  this  expression  by

<*n     "„,

<rn   being  merely   a  prefix    assumed    to   express  a   certain
operation  performed  on  WB.

In  the  same  way   the  first  term   of  F,  viz.

4-Tr  Ip  A0da',

"0

will  evidently  become

4>7r  Ty  {  k  —  +  (l  -  k)  a]  da,

or

4>Tr  J     p  a'2  da  +  4nr  f   l  p  a  da  .
Hence  the  development  of  V  becomes  finally

F  =  —  fa  pa2da'  +  4>7r  /    '  pa  da

a  Jo    r  Ja     (

4-  4-TTca2    cr0w0+  ff\^\  +  cr2w2+  Sue.

It  is  evident  that  this  is  a  series  of  Laplace's  coefficients,
the  sum  of  the  first  three  terms  being  a  coefficient  of  the
order  0,  and  the  succeeding  terms  of  the  order  1,  2,  3,  Sec.
respectively.

41.  We  shall  now  express  the  value  of  Fgot  from  the
equation  (A")  (see  Art.  13)  in  Laplace's  coefficients;  to  do
this  we  observe,  by  trial,  that  -^  —  /m2  is  a  Laplace's  co-
efficient of  the  order  2  ;  hence  the  value  of  V  got  from  (A")
will  be  arranged  in  Laplace's  coefficients  by  simply  putting
it  in  the  form

34

42.  If  we  now  subtract  the  two  equations  we  have
thus  obtained,  we  shall  eliminate  V  and  arrive  at  an  equa-
tion consisting  of  a  series  of  Laplace's  coefficients,  which,
by  (Art.  36),  must  be  put  separately  equal  to  zero;  hence
we  get  the  following  equations,  viz.

—          'o'^d'  '

l      pada'  +  4>Tra)2(r0u0  —  C  +  -  =  0,

a  Jo    '  Ja     '  2

—  (^  -  fjf)  =  0,

and  a-nun  =  0,  for  all  values  of  n  except  0  and  2.

Thus  we  have  eliminated  V  without  knowing  what
function  it  is,  and  obtained  equations  for  determining  w0,
MJ,  uZ9  &c.,  and  therefore  the  equation  to  any  stratum.

43.  We  shall  now  proceed  to  solve  these  equations,
commencing  with  the  last  of  them,  viz.

<rnun  =  O...(l),  except  n=0  or  2.
By  (Art.  40)  this  equation  is  equivalent  to

-^  rapfartda'  +  -  -  l—  -  -  fa  p-^.(a'n+*fun)da
a  Jo    ^  (2n  +  l)o"+l./o    ^  da

a"        ro,   ,   d    (  '«.  \       ,

/    P  -r-7    -r-i  )da  =  °-
1  Jo   r  da   k«*~y

+
2n  +

Now,  by  Note,  p.  24,  if  we  multiply  this  equation  by  aw+1,
and  differentiate  it  relatively  to  a,  and  then  multiply  it

by  —  ,  and  differentiate  it  again  relatively  to  a,  we  shall

by  this  process  get  rid  of  the  two  last  integral  signs,  and
arrive  at  a  differential  equation  which  may  be  put  in  the
form

•   dun
A         +4...0  ......  (2),

35

Alt  Ay  being  functions  of  a  and  p,  which  we  have  no  oc-
casion to  determine.  Now  let  v  and  v'  be  two  quantities
which  satisfy  the  equations

crnv  —  an  =

1  .........  (3),

and  let  C  and  C"  be  two  constants,  then

un  =  Cv  +  CV

will  be  a  solution  of  the  differential  equation  (2)  ;  for,  per-
forming the  operation  an  on  both  sides  of  the  equation,

un  =  Cv  +  CV,
we  have  &nun  =  crn(Cv  +  CV)

=  C(TnV  +  C'(Tnv'

(evidently  from  the  nature  of  the  operation  <rn)
or  ffnun  =  Ca"  +  C'  —^  ,  by  (3).

Now  if  we  get  rid  of  the  integral  signs  in  this  equation  by
the  same  process  we  have  applied  to  the  equation  (1),  the
second  side  will  evidently  be  made  zero  by  the  differenti-
ations, and  thus  we  shall  arrive  at  the  same  differential
equation  as  before;  hence

»,-Ca  +  CV«

is  a  solution  of  (2).  Now  this  is  evidently  true  whatever
be  the  values  of  C  and  C";  hence  this  solution  contains
two  arbitrary  constants,  and  is  therefore  the  most  general
solution  that  (2)  admits  of.  Hence  all  values  of  un  which
satisfy  (l),  since  they  also  satisfy  (2),  must  be  values  of

*  It  is  evident  that  v  and  v'  are  two  different  functions,  otherwise  the
equations  (3)  would  give

which  is  not  the  case.  This  remark  is  necessary,  for  if  v  and  t>'  were  not
different  functions  then  C  and  C'  would  add  together,  and  therefore  be  equi-
valent to  only  one  constant.

Cv  +  C'v.     To  determine  what  values  of  Cv+C'v'  satisfy
(1),  substitute  Cv  +  C'v  for  un  in  (1),  and  we  find

or     Ca"  +  C'  —  =  0,   by  (3).

Now  this  equation  ought  to  be  true  for  all  values  of  a  ;
hence  (7  =  0,  and  C'=0;  hence  it  is  evident  that  only
one  value  of  Cv  +  C'v,  namely  zero,  satisfies  (l);  and
hence  it  follows  from  the  equation  (l),  that

Thus  un  is  zero  for  all  values  of  n  except  0  and  2  ;  this
result  produces  a  considerable  simplification  in  the  equation
to  the  strata.

44.     We  shall  next  consider  the  equation  involving  w2,
which  may  be  written  thus,

In  this  equation  we  may  conceive  w2,  which  is  a  function
of  n,  vi  -  ^,  cos  0,  and  sin  0,  to  be  developed  in  a  series
of  powers  of  i  —  /j?  and  <£.  Let  y  (^  —  yuc)m0"  be  any  term
in  this  development,  -y  being  an  unknown  coefficient  to  be
determined,  then  the  corresponding  term  in  the  equation
will  be

except  m  =  1  and  n  =  0,  in  which  case  it  will  be

Now  i  —  /a?  and  0  are  arbitrary  ;  hence  we  must  put  the
coefficients  of  their  several  powers  separately  equal  to  zero  ;
and  hence  for  all  terms,  except  that  in  which  m  =  1  and
n  =  0,  we  have

0"27  =  °  i  and  therefore  y  =  0,  as  before  in  (Art.  43)  ;

37

and  for  the  term  in  which  m  =  1   and  n  =  0,

a2
ff*V  +  ^.=  \

which  equation  will  determine  y  ;   hence  we  have

w2  =  7  (i  ~  M8)»
where  -y  is  given  by   the  equation

45.  We  cannot  determine  UQ  from  the  remaining  equa-
tion in  Art.  42,  on  account  of  the.  unknown  constant  C  in-
volved in  it  ;  but  the  value  of  u0  follows  from  the  assump-
tion that  we  have  made  respecting  a  in  (Art.  12)  ;  namely,
that  it  equals  the  radius  of  the  sphere  of  the  same  capacity
as  the  stratum  whose  parameter  it  is:  for  this  assumption
gives

—  =>  volume  included  by  stratum,
3

rtir     r  1     /«

I.  Li

a3     rz-T

jjf

a(l+a>8u)

47ra3a>8w0>  by  (Art.  22.)

Hence   UQ  =  0.

46.  Thus  we  have  determined  the  values  of  w0,  M,,
w2,  &c.  ;  and  it  now  only  remains  to  substitute  these  values
in  the  equation,

r  =  a  .  {  I  -i-  or  (MO  +  ut  +  u2  &c.)  |  ;

38

and  we  find  that  the  equation  to  any  stratum  whose  para-
meter is  a,  is

Now  the  equation  to  a  spheroid  generated  by  an  ellipse
revolving  about  its  minor  axis,  is

V  1  -  e2cosa  l^  -  9\

a  being  the  major  axis,  e  the  eccentricity,  and  0  ,the  angle
which  r  makes  with  the  minor  axis.  Supposing  e  very
small,  this  equation  becomes

e2      e~
r  =  a  \l +  ~sin20*

2        2

=  a  {l  -€cos20j,
e  being  the  ellipticity,

maior  axis  —  minor  axis

for  the  ellipticity  =  — ? : :

major  axis

-,  nearly.

This  equation  may  evidently  be  made  to  coincide  with
the  equation  to  the  stratum,  by  putting

a  e  =  o  to2  ;

to2'/
and  therefore    e  =  -  ::  —  =  ft)2/Y»  nearly.

Hence  the  strata  are  all  spheroids  of  revolution  about
the  polar  axis ;  the  ellipticity  of  any  stratum  being  ufy,
y  being  got  from  the  equation
o2  _

STT

or  putting  in  this  equation  —  for  y,  the  equation  for  deter-

(t)

mining  the  ellipticity  of  any  stratum,  will  be

8?r
or,  by  Art.  40,

—  /    p a  da'  H /    p'  — "'  (a'5e')dar

a  Jo  5a  Jo       da

a8    /-"i    ,de  0*0?

+  —        p  — ;  da  + =  0,

5  Ja    ^  da'  STT

which  equation  will  give  e  when  p'  is  known  as  a  function
of  a'.

47.  Thus  we  have  arrived  at  the  remarkable  result
that  the  mass  must  be  arranged  in  strata  of  equal  density,
which  are  all  spheroids  of  revolution  about  the  axis  of  rota-
tion, their  ellipticities  being  connected  by  the  equation  just
obtained.

It  is  evident  that  our  investigation  gives  us  no  infor-
mation respecting  p  ;  hence  the  law  of  density  of  the  strata
is  quite  arbitrary,  and  must  be  determined,  if  possible,  by
some  independent  method.

48.  We  shall  presently  shew  that  the  results  of  our
investigation  may  be  compared  with  observation  in  a  most
satisfactory  manner,  without  knowing  any  thing  of  the  law
of  density  of  the  strata.      This  is  fortunate,  as  we  have  no
means  of  determining  this  law,  but  must  have  recourse  to  an
hypothesis  which  it  must  be  confessed  is  rather  empirical ;
but  as  the  results  it  leads  to  may  be  made  to  agree  well  with

observation,  we  must  look  on  it  as  probable.  The  hypo-
thesis we  allude  to  is  this,  that  the  variations  of  the  pressure
in  the  interior  of  the  earth  (supposed  fluid  and  of  the  same
chemical  constitution  all  through,)  are  proportional,  not  to
the  variations  of  the  density  as  in  gases  of  uniform  tempe-
rature, but  to  the  variations  of  the  square  of  the  density  ;
i.e.  that  instead  of  having  dp  =  kdp,  we  have  dp  =  kpdp.
There  is  some  slight  reason  a  priori  for  assuming  this
formula,  for  it  is  evident  that  p  ought  to  increase  more
rapidly  with  p  in  the  fluid  composing  the  earth,  than  it
would  do  in  gases,  both  on  account  of  the  incompressibility
of  that  fluid,  and  the  increase  of  temperature  as  we  go
towards  the  centre  ;  and  hence  kp  dp  will  represent  the
variations  of  p  better  than  kdp.  But  the  chief  reason  for
assuming  this  formula  is,  that  it  leads  to  correct  results,  and
simplifies  the  equations  we  shall  be  concerned  with,  as  will
appear.

49-  Assuming  then  this  connection  between  the  pres-
sure and  density,  we  may  calculate  the  law  of  density  from
the  equations

....................................  (1),

V  =  ^-J^  p'a'da'+tTT  rip'a'da  .........  (2);

which  are  got  from  (Art.  6)  and  (Art.  42)  ;  neglecting  or,
for  the  centrifugal  force  will  make  a  very  little  difference
in  the  law  of  density,  and  it  will  be  useless  to  be  very
accurate  here,  as  we  are  proceeding  on  rather  uncertain
grounds.

We  have,  from  (l),

dp_         dV
d^**  P  da

multiplying  this  by  —  ,  and  differentiating  relatively  to  a,

P
we  have

d

Substituting  the  assumed  value  of  dp,  viz.  dp  —  kpdp,
we  have

which  may   be  put  in  the  form

Therefore,  putting  —  =  g8,

pa  =  A  sin  (70  +  5),
sin

and  p  =

A  and  B  being  arbitrary  constants.

50.      To  determine  B,   let  a  =  0,  then  we  have
p  =  co  ,  unless  J9  =  0  ;

hence  since  p,  as  we  may  evidently  assume,  is  not  infinite
at  the  centre,  B  =  0,  and  we  have

A  sin  qa
P  ~         a

To  determine  A  and  q,  let  pj  be  the  density  of  the  superficial
parts  of  the  earth,  i.  e.  the  value  of  p  when  a  —  a,  ,
and  let  D  be  the  mean  density  of  the  earth  ;  then

mass  of  earth

its  volume
6

47T

42

/     pa*  da'

JQ

s

3  A

3A     ra,          .

=  — —  I      a  sin  q  a  da  ;
a3  J0

"T     ^  sin  go'

3A

-     .

also      0i  =  —  sin
a,

hence

sin  9^!  —  qal  cos  graj       9  a\  JJ

from  which  equation  q  may  be  found,  and   then  we  shall
have  A  from  the  equation

sin  9  a,

51.      Observation    shews    that   —  is  about  — ,  and  on

D  11

substituting  this  value  of  --j-  in  the  equation  for  deter-
mining (7,  we  shall  find  by  repeated  trials  (which  is  the
only  way  we  can  solve  it)  that  it  admits  of  several  solu-
tions, of  which  one  only  leads  to  right  results;  it  is  this,

1       5-7T      .

hence,  substituting  —   —  for  (7,  we  have

.    .     /57T    a
A  sin  |  —    —
o     a.

43

To  determine  A,  we  shall  put  a  =  a,  in  this,  which  gives

A     .     5-n-

«  sin  T

-  sin  30°

and  hence  .4  =  S^e^,  and  we  have

O

which,  if  our  hypotheses  be  correct,  expresses  the  density
of  any  stratum  in  terms  of  the  parameter  of  that  stratum,
and  the  superficial  density.

52.  The  method  by    which   we  have   arrived   at   this
formula   for  the   density  is  not  very   satisfactory,  and   we
shall  therefore  consider  it  as  empirical  ;   we  observe  that  it
gives  a  density  which  increases  as  we  go  towards  the  centre,
but  does  not  become  infinite  there;  this  is  most  probably
the  case;   it  also  makes  the  pressure  vary  more  rapidly  as
we  approach  the  centre  than  it  would  do  if  the  earth  were
gaseous    and    of  uniform   temperature;    this   is   also   most
probably  the  case;    and  it  gives   the  mean  density  of  the
earth  its  proper   value:   we   shall  prove  presently   that   it
gives  the  true  value  of  the  earth's  ellipticity,  and  also  the
true  value  of  the  coefficient  of  precession;   hence,  on  the
whole,  we  may  assume  it  with  some  probability  as  the  law
of  density.

53.  Hence,  finally,  it  follows  from  the  hypothesis  of
the  earths  original  fluidity  ;

(1)  That  the  earth  ought  to  consist  of  equidense
strata,  all  spheroids  of  revolution  about  the  axis  of
rotation.

44

(2)     That  if  e  be  the  ellipticity  of  any  of  these  strata
it  satisfies  the  equation

-  i/v«w  +  A/V/--  <••  vv°'

a  Jo   r  5a?Jo   r  da

a2    /">i  ,de    .    ,     a2
'

a     /">i  ,e    .    ,     a  <t)
+  —   /     p  —  ,  da'+  -  =  0.
5  >    r  da  STT

(3)     That  we  may  assume  with  probability  the  law  of
density  to  be

a  /STT    a\

p  =  2/0,  -  sin     —    - )  .
ri  at          \  6     aj

CHAPTER   IV.

METHODS    OF     COMPARING    THE     RESULTS    JUST    ARRIVED
AT    WITH    OBSERVATION.

54.  IN  order  to  test  the  correctness  of  the  conclusions
we  have  just  arrived  at,  we  shall  now  deduce  from  them
results  of  a  more  practical  character,  which  shall  admit  of
direct  comparison  with  observation  :  the  first  result  we  shall
deduce  is  this  ;

If  *  be  the  length  of  a  meridian  arc,  measured  from  the
pole  to  any  place  whose  colatitude  is  c,  then

a  and  ey  being  the  major  axis  and  ellipticity  of  the  earth.
For,  by  Art.  48,  the  equation  to  any  meridian  is
r  =  a  (1  -  e,  cos2  0)

=  «(l--cos20)  .........  (1).

XT  ^^

Now_=          r*

=  r,  neglecting  squares  &c.  of  e,

_«{!-!.<_  |  COg  20}     by    (,),

therefore,  integrating,

(2).

46

Adding  no  constant  because  «  evidently  =  0  when
0  =  0.

Now  c  (the  colatitude)  is  the  angle  made  by  the  normal
with  the  polar  axis  ;  hence  (9  —  c)  is  the  angle  made  by  the
normal  and  radius  vector,  and  hence

or,   neglecting  the  squares  &c.  of  small  quantities,
9-c  =  e  sin  20,    by  (1),

therefore

9  =  c  +  e,  sin  2  9
=  c  +  et  sin  2  (c  +  e  sin  20)
=  c  +  e/  sin  2c  nearly  ;

hence,  substituting  in  (2),  we  have

a  \c  +  e  sin  2c  --  -  c  —  -  sin  2c|,
2  4

55.  To  shew  how  this  result  may  be  compared  with
observation,  let  s'  and  c'  be  the  values  of  s  and  c  correspond-
ing to  another  place  near  the  former,  and  on  the  same
meridian,  then

and  therefore
*'  -  s  =  a

j  1  1  -  |)  (c'  -  c)  +  ^  (sin  2c'  -  sin  2c)|  ,

cos    ''+  c>   sin    c'-6''

47

Now  in  this  equation  s'  —  s,  being  the  distance  between
two  places  near  each  other,  may  be  determined  by  the  usual
method  of  triangulation  ;  cos  (c  +  c)  may  be  found  by  any
of  the  ordinary  methods  of  determining  the  latitude  of
places,  without  aiming  at  any  great  accuracy,  since  it  is
multiplied  by  the  small  quantity  e  ;  c'  —  c,  not  being  mul-
tiplied by  a  small  quantity,  must  be  determined  more  ac-
curately by  observing  the  meridian  zenith  distances  of  the
same  star  at  the  two  places,  and  taking  the  difference
which  will  evidently  be  equal  to  c  —  c  ;  thus  we  may  put
our  equation  in  the  form

when  A,  B,  C  are  known  '  quantities  got  by  observation.

In  the  same  way,  by  observations  at  other  places,  we
may  obtain  any  number  of  similar  equations;  suppose
them  to  be

&c  .......  &c  .......

From  any  two  of  these  equations  we  may  determine
a  and  e,  ;  and  the  values  of  a  and  e  so  determined*  ought  to
satisfy  all  the  other  equations;  hence,  if  we  find  that  all
these  equations  are  satisfied  by  the  same  values  of  a  and  ey,
it  is  evident  that  our  result  agrees  with  observation.

56.  Now  a  number  of  meridian  arcs  have  been  mea-
sured, and  a  system  of  equations  similar  to  the  above  have
been  formed,  and  it  is  found  that  the  values

a  =  3.Q62.82  miles,   e  =  —  ,

'      306

satisfy  them  all  to  a  remarkable  degree  of  accuracy,  allowing
for  certain  small  errors  which  may  be  easily  accounted  for  ;
and  which,  even  considering  them  in  the  most  unfavourable
point  of  view,  are  very  much  smaller  than  e,,  which  is  itself

*  Or  rather,  the  values  of  a  and  e,  got  from  all  the  equations  by  the  method

of  least  squares.

48

a  very  small  quantity ;  and  indeed  if  we  bear  in  mind  the
delicacy  and  number  of  the  observations  requisite  in  order
to  form  the  above  equations,  the  smallness  of  the  errors
is  most  remarkable.

57.  On  the  whole  we  are  justified  in  concluding  from
observation,  that  the  equation  to  any  meridian  and  there-
fore to  the  earth's  surface  is  very  nearly  this,  viz.

r  =  a  {  1  -e/cos80},
where  a  =  3902.82  miles,

and  €  —  —  .
'      306

Hence  the  hypothesis  of  the  earth's  original  fluidity
leads  to  a  very  peculiar  result,  which  is  capable  of  varied
and  extensive  comparison  with  observation,  and  which
agrees  with  it  in  a  remarkable  manner;  from  which  we
must  conclude  that  this  hypothesis  is  most  probably  true.

58.  The    second    result    we    shall    deduce    from    our
theory  is  this  ;

If  g  be  the  force  of  gravity  at  any  place  whose  colati-
tude  is  c,  then

g  =  U  [  1  +   I   — 61   I  COS'C^  .

Where  G  is  a  constant,  namely,  the  value  of  g  at  the
equator,  and  m  the  ratio  of  the  centrifugal  force  to  gravity
at  the  equator*.

*  To  determine  w,  we  observe  that

G  -  32.2  +  a  small  quantity,
hence

ura

32.2  +  a  small  quantity

In  putting  32.2  for  G  we  have  assumed  a  foot  to  be  the  unit  of  length,
and  a  second  the  unit  of  time;  hence  we  must  express  a  and  to2  in  terms  of
these  units,  and  therefore  we  have

a  =  3962  miles,  nearly,

=  3962  x  5280  feet,

and

49

To  prove  this,  we  observe  that  -g  is  the  resultant
of  the  forces  which  act  on  a  particle  at  the  surface,  which
forces  are,  by  (6),

dV  dV  dV

—  +  ftT<r,    —  -  +  ory,  —  —  ,
das  dy  dz

and  therefore  g  must  balance  these  forces  ;  hence,  by  the
principle  of  virtual  velocities,  if  dr  be  any  variation  of
r,  and  da?,  dy,  d#,  its  resolved  parts  along  the  axes  of
co-ordinates,  and  >//  the  angle  which  r  makes  with  the
direction  of  g,  we  shall  have

.r  )  dx  +  (  --  \-  u?y  }dy  +  —  dsr,
\dy  /    *      dx

which  gives

I
\d.T.

"

r

Now  g  acts  in  the  normal  (by  the  principles  of
Hydrostatics),  and  the  normal  evidently  makes  a  very
small  angle  with  the  radius  vector  ;  hence  \|/-  is  very  small,

V

and  therefore,  since  cos  \J/  =  1  -  —  +  &c.,   we  may,  neglect-

ing  squares   and    higher   powers  of  small  quantities,  put
cos  \js  =  1  ;   moreover  we  have,  by  (6)  and   (42),

and

4-Tr  r<i   ,  ,n  ,  ,  /*",    ,   .  ,  ,      w2a2

C  =  —  /    pa*  da  +  4-TT        p  a  da  +  -  ;
a  Jn  f  Ja    '  3

Htld    ">  =  24ho^
2-r

24x60x60'

Mnkins;  the<:e  substitution*,  we  shall  find
m  =  about  — —  *

50

hence,  observing  that  a  is  a  function  of  r  in  virtue  of  the
equation

r  =  «{l+e(i-M2)},
we  have,   substituting  in   (1),

dC

g=-dr
_dCda
dr  dr
*TT   ra  ,  ,     2ft,2  a\J          d(ae)  \

a  da  —   l  ~  -     (-M

which,    neglecting   the   product   of   small  quantities,  may
evidently  be  put  in  the  form

where

G  **  -—    I   p'a'ada'  +  small  quantities.

Since  g  *=  G  when  /u  =  0,  it  is  evident  that  G  is  the
force  of  gravity  at  the  equator.

(Of  course,  in  all  our  formulae,  at  is  supposed  to  be
put  for  a  after  all  differentiations  and  integrations  have
been  performed,  since  the  particle  on  which  g  is  the  force
is  supposed  to  be  at  the  surface.)

Now  by  (53),

-1  (ap'a'zda'+~  ^  o'-^
a  Jo   P  5a5  Jo   r  da

a2    /*«i    ,de    ,  ,
+  —   /      P  -r-,  ;  da  +  -  =  0.
5  Ja     ?  da  STT

Multiplying  this  equation  by  a3,  and  differentiating  relative-
ly to  a,  we  have

d(ea?)    ra   ,          ,          ra,   ,de        ,     5ft,2  a4

-    /    oVW+a4  /     o—  ,da'+-    -  =  0,
da      Jo    r  J"    r  da  STT

51

or,  observing  that  af  is  to  be  put  for  a,

8?r  /    p  a>  da
_  5g,V

=  TcT'

since  G  =  ~  f"  pa'*da  +  small  quantities,

fl       ''O

5ma

= 5

centrifugal  force        a>2o

since  m  =  ; =  — - ;

gravity  at  equator        G

hence   — = —  -  e

da          a     da

5m
and  hence  equation  (l)  becomes

since  0  =  c  +  esin2c,  by  (64).

59-     To  shew  how  to  test  this  result  by  observation,
we  observe  that  if  p  be  the  length  of  the  seconds  pendulum,

then  since  1  =  27T  \/-  ,  we  have  p  =  ^L

x-i

when  P  =  — 2  =  value  of  j»  at  the  equator.

52

Now  p  may  be  determined  by  observation  at  any  place ;
hence,  by  observations  at  different  places,  we  may  find  a
system  of  equations  such  as  before ;  viz.

See.  &c.

when  Ay  B,  A',  B">  SEC....  are  quantities  got  from  observation.
Now  it  is  found  that  the  values

,  5m
P  =  39.01228  inches,  and   --  e/  =  .005321,

satisfy  all  these  equations,  not  so  exactly  as  before,  but  yet
with  remarkable  accuracy,  considering  the  small  quantities
we  are  engaged  with;  hence  this  result  is  another  proof
of  the  Earth's  original  fluidity.

60.     The  comparison  of  this  result  with  the  former  is
a  strong  additional  proof;  for  the  former  result  gives

and  since  we  know  that  m  =  -  ,  the  present  result  gives

this  is  a  very  remarkable  coincidence,  and  must  be  consider-
ed as  a  decided  proof  of  the  correctness  of  our  hypothesis.

61 .     If  G'  be  the  value  of  g  at  the  pole,
G'=G

_
and

53
G'-  G      5m

/

This  result  is  Clairaut's  theorem.

62.    We  may  determine  the  Earth's  ellipticity  by  means
of  the  law  of  density  assumed  in  Chap.  in.

The  equation  for  determining  e  is

-  1  f'p'aPda'  +  ~  f  V  J~,  (ea*)daf
aJ0  r  5asJ0  r  da

a     /•«,   ,de'       ,     «r  a*
+  -/     P^-7da'+  --  =  0.
5  Ju    r  da  8-rr

Integrating  the   second    and  third  terms  by  parts,  we
have

a2  r^du    ,  ,  ,
—

a    r^u    ,  ,  ,      of  a
--  /      —  ,  e  da  +  L-     -
5  Ja     da  5

5  Ja     da  5  8?r

A  sin  qa
Now  putting  p  =  —         —  ,  we  have

/     p'a'~da'=  —  (sin  qa  —  qa  cos  qa)

dp  A  ,

-1-7  =  --  -  (sin  q  a  -  qa  cos  qa  )  ;

hence,  if  we  assume

e  (sin  qa  -  qa  cos  qa)  =  »;,
(1)  will  become

At)        A      ra     ,    t        ,      A  a*   /-a,  1J.    .    ,        ,  O€.       (O*\    ,

--5-+—  ,/    via*dc!+—  -       -±da+(^+  —  a2  =  0;
(f-  a     5aiJo  5   Ja    a'*  \  5        S-rrJ

dividing  this  by  a2,  and  differentiating  relatively  to  a,  we
have

A   d    /  ri\        A   ra  ,          ,
-  -  —  [-J    -  -  /    na    da  '=  0  ;
tfda\\arj       erJ*

54
or  multiplying  by  a6,  and  differentiating  again,

63.      To  solve  this  differential  equation  we  shall  assume
£'  such  a  function  of  a  that

therefore

Hence,  substituting  in  the  equation  (2),  we  have
a??-3a  fVr'da'  +  (fa  /V  Fa'?  da'*  ;

»/0  ''O          •'O

or  dividing  by  a  and  differentiating,

*g-«i;+  *".f  *?'«•:

or  again  dividing  by  a  and  differentiating,

hence    <£=  C  sin  (qa  +  C'),     C    and    C'    being    arbitrary
constants,  and

r  a'^da  =  —  {sin  (qa  +  C')  -qacos(qa  +  C')},
and

{"a    faa<gda'  =  —  {3  [sin  (qa  +  C')  -  qa  cos  (qa  +  C')]

55

Hence  we  shall  have,    putting  C   instead  of  —  ,
rj  =  C  |  -y-j  [sin  (qa  +  C')-qa  cos  (qa  +  C")]  -  sin  (qa  +  C')>  .

64.  We  might  determine  the  constants  C  and  C'  by
substituting  this  value  of  rj  in  (l),  Art.  62,  but  the  following
method  will  be  more  simple  :  in  the  first  place  we  may  see,
a  priori,  that  C'  must  =  0,  for  otherwise  we  should  have
»/,  and  therefore  e  very  large  when  a  is  very  small,  contrary
to  our  assumption  of  the  nearly  spherical  form  of  the  strata  ;
hence

rj  «•  CJ-jj  (sinqa  -  qacosqa)-  sin  qa  >  ......  (2),

and  therefore,  since  rj  =  e  (sin  qa  —  qacosqaj,

.  --

q2a2      sin  qa  -  qa  cos  qa)

i  s        2\

We  shall  determine  C  by  means  of  the  value  of     ^ea  '  ,

da

got  in  p.  51.

Multiplying  (2)  by  a2,  and  differentiating  relatively  to
a,  we  have

d(r,a2)

— =  Ca  (sin  qa  —  qa  cos  qa),

da

also  doing  the  same  to  the  equation

tj  =  e  (sin  qa—  70, cos  qa),
we  have

— ; =  — i (sin  qa  -  qa  cos  qa)  -f  eg2 a3  sin  qa.

da  da

Equating  these  values  of ,   we  have

da

1   d  (ea8)  sin  qa

'+^0*^

oa  sin  ga  -  qa  cos  70

56

.  d(ea3)       5  ma
Hence  putting  a  =  a  ,  and  ---  -  =  -  •*  ,  see  p.  51,

5m      36/p,

=  TH  -~D~'

also  putting  a  =  nf  in  (3),  we  have  by  Art.  50,

from  which  two  equations  we  get
5m

"i"

»H-  D

XT  1  5ir     p          5

rsow  m  = ,  qa  =  — ,   —  =  — ,

289          '       67)      11

substituting  these  values  and  reducing,  we  find

306

This  result  agrees  with  observation,  but  the  agreement
is  not  of  much  value  on  account  of  the  assumption  of  the  law
of  density.

65.  The  effect  which  the  attraction  of  the  Earth  has
on  the  Moon's  motion,  is  usually  brought  forward  as  an-
other means  whereby  we  may  test  the  correctness  of  the
hypothesis  of  original  fluidity ;  and  the  agreement  between
theory  and  observation  in  this  particular  is  considered  to
afford  additional  evidence  of  the  truth  of  the  hypothesis.
We  shall  attempt  however  to  prove  that  this  is  not  the  case.
To  do  this,  we  shall  shew  that  if  the  equation  to  the  Earth's
surface  be  known,  and  also  the  law  of  variation  of  the  force
of  gravity,  then  the  effect  of  the  Earth's  attraction  on  the
Moon  follows  as  a  necessary  consequence,  independently
of  any  theory  except  that  of  universal  gravitation.

57

66.  It  is  evident  from  the  smallness  of  the  variations
of  the  force  of  gravity,  that  the  Earth  must  consist  of
nearly  spherical  strata;  hence  all  the  results  we  have  al-
ready obtained,  so  far  as  they  depend  on  the  nearly  spheri-
cal form  of  the  strata,  will  be  true  whether  the  hypothesis
of  original  fluidity  be  correct  or  not.

Hence,  as  before  in  p.  49,  we  shall  have

putting  a/  for  r  in  the  small  term.

*  Now  the  expression  for  V  in  Art.  40.  may  evidently  be
put  in  the  form

a  series  whose  general  term  is

+  47TO)2

,   d

hence,  differentiating  relatively  to  r,  and  observing  that  a  is
a  function  of  r  in  virtue  of  the  equation  r  =  a  (l  +  o>2w),  and
supposing  that  at  is  put  for  a  after  the  differentiations,  we
have

flTF__47r    /.Ol    ,  ,      /47r         „  \  da

~a7~~^J0    pa

*  For  the  part  of  V  which  is  multiplied  by  4ir  £p'a'eda'  is  evidently
,&c.)},  which  =-.

58
a  series  whose  general  term  is

(  a  series  wnose  general  term   is  \

1  (da

<.  n  +  1          ra,       d  „       ,    ,    ,  ,  >    ,

\--r  -        -=  /     P  —  ,(una*+3)da  +  pauA  dr
I      2n+l«n+2  Jo    r  da'v  '  '  '  ")

,

+  47ro>z

-

(2n+l)«<

Now  since  r  =  a  (l  +  ct>2w),

c?a
and  therefore  —  =1  -  &>

da
we  have,  evidently,  neglecting  o>4  &c.,

MTT  \  da  _  f

\rt  ?'  '  *  ')  dr  P'  '

and  therefore,  neglecting  a>4  &c.,

4£--?r>w

a  series  whose  general  terra  is

47TO)2

hence  if,  for  brevity,  we  put

shall  have

-o>2ai  (1  -/x2);
and  hence

47Tft)2  ,  2  3

— 5-  { za  +  -  z,  +  -5  z2  +  &C.  5

59

Now  if  we  suppose  rj  and  g  known,  it  is  evident  that
the  first  member  of  this  equation  will  be  known,  and  may
therefore  be  supposed  to  be  expanded  in  a  series  of  known
Laplace's  coefficients ;  and  hence,  since  the  Laplace's  co-
efficients of  different  orders  on  each  side  of  the  equation
must  be  separately  equal,  by  Art.  36,  the  values  of  Z0  Zl  Z2
&c.  will  be  known.

*  Now  it  is  evident  immediately,  from  Arts.  14.  and  40,
that  the  value  of  V  for  any  external  point  is

V=  —  faip'a'*da  +  47rte)2(-  Z0  +  - -  Z,  +  4  Z2  +  &c
r  Jo  [T  r  r

hence,  since  Z0  Zt  Z2  SEC.  are  known,  as  we  have  just
proved,  it  is  evident  that  the  value  of  V  for  any  external
point  is  also  known.

67.  Hence,  if  we  know  the  form  of  the  Earth's  sur-
face, and  the  law  of  the  variation  of  gravity,  we  shall
know  the  value  of  V  for  any  external  point,  and  there-
fore be  able  to  determine  the  attractions  of  the  Earth  on
that  point  without  making  use  of  the  hypothesis  of  original
fluidity.

Hence  it  follows  that  if  the  form  of  the  Earth's  surface
and  the  law  of  variation  of  the  force  of  gravity,  calculated
on  the  hypothesis  of  original  fluidity,  agree  with  observa-
tion, then  the  effect  of  the  Earth's  attraction  on  the  motion
of  any  external  body,  such  as  the  Moon,  calculated  on  the
same  hypothesis,  must  also  agree  with  observation,  whether
that  hypothesis  be  true  or  not;  and  hence  we  conclude,
that  the  motion  of  the  Moon  does  not  afford  additional
evidence  of  the  Earth's  original  fluidity.

*  For  the  only  difference  made  in  the  reasoning  in  Art.  40,  by  using  the
expression   for  V  given  in  Art.  14  instead  of  that  given  in  Art.  15,  will  be

simply  this,  that  we  shall  have  to  consider  —  instead  of  — ;  also,  since  the  attract-
ed  point  is  external,  and  therefore  r  always  greater  than  a',  it  is  evident  that  k
will  be  always  unity.

In  the  next  chapter  we  shall  determine  the  equations
of  motion  of  a  rigid  body  round  its  centre  of  gravity,  and
thence  deduce  the  Earth's  motion  round  its  centre  of  gravity.
We  shall  find  that  the  result  affords  a  confirmation  of  the
law  of  density  assumed  in  Chapter  in.,  and  also  of  the
hypothesis  of  original  fluidity.

CHAPTER  V.

EQUATIONS    OF    MOTION    OF    A    RIGID    BODY    ROUND    ITS
CENTRE    OF    GRAVITY.

68.  WE  know  from  the  principles  of  Dynamics,  that
a  rigid  body  acted  on  by  any  forces  moves  relatively  to
its  centre  of  gravity,  in  the  same  manner  as  if  that  point
became  fixed,  all  other  dynamical  circumstances  remaining
unaltered;  hence,  whenever  we  wish  to  investigate  the
motion  of  a  body  relatively  to  its  centre  of  gravity,  we
may  consider  that  point  as  fixed,  and  this  will  render  the
investigation  simpler.

Suppose,  then,  that  we  have  a  body  whose  centre  of
gravity  is  fixed,  acted  on  by  any  forces ;  let  §m  be  any  ele-
ment of  it,  any  %  the  co-ordinates  of  Sm  at  the  time  t  referred
to  any  arbitrary  rectangular  axes  fixed  in  space,  and  origi-
nating in  the  centre  of  gravity  ;  let  L  M  N  be  the  moments
of  the  impressed  forces  round  the  axes  of  a?  y  %  respectively,
then  we  have,  by  the  principles  of  Dynamics,

tfco

— -  *  -  -—  «  U  M

(A).

69.  In  order  to  perform  the  integrations  denoted  by
2  in  these  equations,  we  shall  introduce  new  variables
instead  of  xyz,  which  shall  have  reference  merely  to  the-

position  and  motion  of  the  whole  body,  and  not  to  any
particular  particle  of  it.  To  do  this,

Let  x  y'  %   be  the  co-ordinates  of  \$m  referred   to  any
arbitrary  rectangular  axes  fixed  in  the  body,  then  we  have
x  =  se  cos  (x x)  +  y  cos  (y'  x)  +  %  cos  (%  x)  ;

differentiating  this  relatively  to  #,  and  observing  that  x  y  %'
do  not  vary  with  #,  we  have

dot       ,  d  cos  (a?' a?)        ,d  cos  (y' x)       ,  d  cos  (z'x)

dt=* — dT-~  +  y~— dT~ +%~~dt — '

Now  the  axes  of  x'  y  %  are  perfectly  arbitrary ;  we
may  therefore  suppose  them  so  chosen  that  they  shall  coin-
cide with  the  axes  of  x  y  %  at  any  instant  we  please.
Suppose  therefore  that  this  coincidence  takes  place  at  the
time  #,  then  we  shall  have  x'  =  x,  y  =  y,  %  =  % ;  and  if,.

„  d  cos  (x'x)     d  cos  (y'x)

for  brevity,  we  denote  the  values  of -^ — - ,    -          y— - ,.

at  at

~ — -,   at  the  instant  of  coincidence  by  X  X'  X"  re-

dt

spectively,  our  equation  becomes

dx

—  =  \x  +  \y  +  \  #;
at

X  X'  X"  are  evidently  variables  which  have  reference  to  the
position  and  motion  of  the  whole  body,  and  not  to  any
particular  particle  of  it,  for  they  depend  simply  on  the
angles  which  the  two  systems  of  co-ordinate  axes  make
with  each  other  at  any  time,  or  rather  upon  the  rate  at
which  these  angles  are  varying  at  the  instant  of  coincidence
of  the  axes.

In  the  same  way  we  may  prove  that

dy

-j~t  =  M  +  ft  %  +  fji  'x,

dss  „

— -  =  vss  +  i >x  4  v  y;
at

where  /M  //  n",  v  v  v"  are  quantities  similar  to  X  X'  X".

63

Now  since  \$m  is  rigidly  connected  with  the  origin,  we
have

xz  +  if  +  s?  =  constant,

dx         dy         dx

and  therefore  x—  +y—  +  ss— =0;
dt         dt         dt

„  dx    dy    dz

substituting  in    this    equation   the   values   of  —    -~-    —

dt    dt     dt

just  found,  we  have

Xa?2  +  fiy2  +  v%*  +  (X'-f  fj.")xy+  (p  +v")yz  +  (v  +  X'')#a?  =  0;

hence,  since  xyz  are  arbitrary,  we  have

X'  +  JM"  =  0      //  +  v"  =  0      v'  +  \"  =  0.

.  dx    dy    dss  ,

Hence  the  values  of  -p-    —    —   become
dt    dt    dt

-
dy

dx

—  =vy-Xx.

To  conform  to  a  common  and  convenient  notation,  we
shall  put  o>j  ft>2  0)3  instead  v"  X"  /^L"  respectively,  and  write
these  equations  thus,

dx

—  .„,»-«*

—  -  =  W3ar-tt,2f
at

we  shall  presently  determine  what  ojj  w2  o>3  are.

These  equations  express  the  relations  which  exist  be-
tween the  velocities  of  any  element  of  the  body  and  its  co-

64

ordinates  at  the  time  £,  in  consequence  of  the  rigidity  of  the

body.       The  substitution  of  these  values   of  —    —    —

at    at    at

in  the  equations  (A)  will  be  very  advantageous,  since
o>i  c«2  ft>3  are  independent  of  x  y  z,  and  may  therefore  be
brought  outside  the  integral  sign  2.  To  perform  this
substitution  we  have  from  the  equations  (J5),  differentiating

dx  dy   dz    .

and  m  the  result  putting  for  —  —  —  their  values  given
dt   dt   dt

by  the  equations  (J5),

d<o3

Now  the  axes  of  so  y  %  are  perfectly  arbitrary,  we  may  there-
fore choose  them  so  that  the  principal  axes  of  the  body
shall  coincide  with  them  at  the  time  £,  and  therefore  we
shall  have  at  the  time  #,

and  hence,  from  the  equations  just  obtained,  we  have  at
the  time  #

at

dt          dt
and  hence  the  first  of  the  equations  (A)  will  become

~^m  (a?  +  f)  +  w^  2\$m  (*  -  y*)  =  N.

Now  the  principal  axes  coincide  with  the  axes  a?  y  z  at
the  time  t,  hence  if  A  B  C  be  the  principal  moments  of
inertia  of  the  body,  we  shall  have

65

23m  (x*  -  y2)  =

(y*  +  *)

and  hence    ^.  —  —  \-  (B  —  A)

AT.

We  may  transform  the  other  two  of  the  equations  (A)
in  exactly  the  same  manner,  hence,  instead  of  the  equations
(-4),  we  have  the  following  three  equations,  in  which  the
integral  sign  2  is  got  rid  of,  viz.

A  —  +  (C  -  B)  ft>2o>3  =  L
at

at

70.  It  remains  to  determine  what  wl  w2  w3  are;  the
equations  (B)  will  enable  us  to  do  this  immediately,  for
from  these  equations  we  find  (putting  %  =  0)  that  the  ve-
locities parallel  to  the  axes  of  x  and  y  of  any  particle  in  the
plane  of  any  are

daa

dy

-=      «,,,*;

and  hence  the  whole  resolved  velocity  in  the  plane  of  xy  of
that  particle,  viz.

will  be  o)3  <\/a?  +  y2.

Now  if  we  suppose  the  particle  to  be  at  a  distance  unity
from  the  origin,  and  therefore  a?2  +  y2  =  1,  this  velocity  will
become  a>3,  and  hence  w3  is  the  resolved  velocity  in  the  plane
of  xy  of  any  particle  situated  in  that  plane,  at  a  distance
9

66

unity  from  the  origin.  Since  the  particle  is  rigidly  con-
nected with  the  origin,  it  is  evident  that  this  velocity  takes
place  perpendicularly  to  the  line  joining  the  particle  and  the

origin ;  and  also  since  —  is  negative,  it  is  evident  that  this
dt

velocity  tends  to  move  the  particle  from  the  axis  of  x  to-
wards that  of  y,

In  the  same  way  it  may  be  proved  that  MI  and  <o2  are
similar  velocities  with  respect  to  the  planes  of  yz  and  zx,
respectively ;  o>i  tending  from  the  axis  of  y  towards  that  of
#,  and  o>2  tending  from  the  axis  of  %  towards  that  of  sc.

71-  To  give  clear  ideas,  we  shall  represent  the  manner
in  which  these  velocities  tend,  by  means  of  the  following
figure.

Let  XY,  FZ,  ZX  be  the  intersections  of  the  planes
of  acy,  yz,  »#,  respectively,  at  the  time  #,  with  a  sphere
fixed  in  space,  unity  being  its  radius,  and  the  origin  its
center ;  then  <a3  will  be  the  resolved  velocity  along  the  great
circle  XY  of  any  point  P  situated  on  that  great  circle,  and
o>j  will  be  the  resolved  velocity  along  the  great  circle  FZ
of  any  point  Q  situated  on  that  great  circle,  and  o>2  will
be  the  resolved  velocity  along  the  great  circle  ZX  of  any
point  R  on  that  great  circle;  and  these  velocities  tend  in
the  directions  represented  by  the  arrows.

67

It  is  evident  from  this,  that  wj,  o>2>  «>3  are  also  the
angular  velocities  of  the  planes  of  yz,  ##,  and  ay,  round
the  axes  of  #,  y,  and  *,  respectively  :  but  it  may  be  easily
seen  from  the  equations  (.6),  that  wlt  w25  <Ws  are  not  tne
angular  velocities  round  the  axes  of  #,  y,  ar,  of  any  other
points  of  the  body,  except  those  situated  in  the  respective
co-ordinate  planes.

72.  We  may  determine  the  position  and  motion  of  the
body  by  means  of  these  velocities,  as  follows.

Let  A  be  any  fixed  point  on  the  surface  of  the  sphere
described  before  (see  the  figure),  and  AB  any  fixed  great
circle;  draw  the  great  circle  AZC  to  meet  YX  produced
in  C.

Let  the  angle  ABZ  =  \^,  the  angle  A  Z  =  0,  and  the
angle  C  Z  X  =  <p;  then  it  is  evident  that  these  angles  define
completely  the  position  of  the  body  in  space  at  the  time  t,
and  may  be  considered  as  the  co-ordinates  of  the  position
of  the  body,  \^  and  0  being  the  co-ordinates  of  the  point  Z,
which  we  shall  call  the  pole  of  the  body,  and  (p  the  addi-
tional co-ordinate  requisite  to  determine  the  position  of  the
plane  of  ZX,  and  therefore  of  the  whole  body;  we  may
easily  determine  these  co-ordinates  from  the  quantities
«!,  w2,  fc>3,  as  follows.

It  is  evident  that  the  velocities  of  the  point  of  the  body
coinciding  with  Z  are,

,  perpendicular  to  A  Z,
o/  1

J  f\

and  —  along  ^Z,

and  the  velocity  along  CX  of  the  point  of  the  body  co-
inciding with  C  is

d\l/

--|  sin  AC,  due  to  the  variation  of  >Jr,

+  -T"5*0  ZC*»  due  to  the  variation  of  0;

68

i.  e.,  since  ZC  =  90°,

Now  by  what  we  have  proved  in  (71),  the  velocities  of
the  body  coinciding  with   Z  are,

&>2  along  ZJT,
and     -  wl  along  Z  F,

and  the  velocity  along  C  X  of  the  point  coinciding  with  C
is  o>3 ;  hence,  since  these  two  sets  of  velocities  must  be
equivalent,  we  have,  resolving  the  latter  set  so  as  to  make
them  coincide  with  the  former,

-J-  sin  9  =  —  MI  cos  0  +  o>2  sin  <
at

—  =  ojj  sin  0  +  o)2  cos  0

(C).

73.  These  differential  equations  connect  0,  \^,  and  6
with  «i,  &)2»  o>3,  and  these,  along  with  the  three  others  (A')
in  (69),  which  connect  o^,  w2,  w3  with  the  impressed  forces,
form  a  system  of  six  equations  connecting  the  six  unknown
quantities  o)15  w2,  ft>3,  \^,  ^,  0  with  #,  and  which  therefore,
when  it  is  possible  to  do  so,  will  enable  us  to  solve  any
problem  respecting  the  motion  of  a  rigid  body  about  its
centre  of  gravity.

The  equations  (B)  will  enable  us  to  determine  the  mo-
tion of  any  point  we  please  of  the  body,  should  it  be
requisite  to  do  so.

74.  These  equations  will  be  sufficient  for  our  present
purpose,  namely,  the  determination  of  the  Earth's  motion
about  its  centre  of  gravity ;  we  shall  hereafter  recur  to
this  subject,  and  deduce  several  interesting  consequences
from  these  equations.

CHAPTER   VI.

PRECESSION    AND    NUTATION.

75.  WE  shall  now  make  use  of  the  equations  deduced
in  the  last  Chapter,  to  determine  the  motion  of  the  Earth
round  its  center  of  gravity.

The  forces  which  act  on  the  Earth,  are  the  attractions
of  the  Sun,  Moon,  and  other  planetary  bodies;  but  on
account  of  the  Earth's  nearly  spherical  form,  the  motions  of
the  Earth  round  its  center  of  gravity,  produced  by  these
forces,  are  but  small ;  hence,  by  the  principle  of  the  super-
position of  small  motions,  we  may  consider,  separately  and
by  itself,  the  effect  of  the  attraction  of  each  planetary  body
on  the  motion  of  the  Earth  round  its  center  of  gravity ;  we
shall  accordingly  commence  with  the  Sun's  effect.

76.  We  shall  first  prove,  that   the  attractions  of  the
Sun  on  any  particle  of  the  Earth  are  the  same  very  nearly
as   if  the   Sun's  mass   were   condensed  into   his  center   of
gravity.

Take  the  center  of  gravity  of  the  Sun  as  origin;  let
xy'%  be  the  co-ordinates  of  any  element  §m  of  it,  and  let
xyz  be  the  co-ordinates  of  the  attracted  particle  §m  of  the
Earth.  9

Then,  if  V  denote  the  sum  of  each  element  of  the  Sun,
divided  by  its  distance  from  \$m,  the  attractions  of  the  Sun
on  \$m  will  be

dV      dV      dV
dx"     dy*     d%'

70

\

Now     V  =  2  —

2  (o?^  +  yy  +  #*')      r'

putting  r  and    r'    for  <#2  +  y2  4-  #2,  and   X*  4-  y'2  +  #'2,    re-
spectively

a?<j?'  +  yy  +  %%  \

r  }

expanding  and  neglecting  the  squares,  &c.  of  the  very  small

x'     11      %'          ,  r

quantities   — ,  — ,  — ,    and   -

r      r      r  r

1

=  —  2owi .
r

Since  2^m,t?',  25my',  2^m^'  are  zero,  because  the  origin  is

center  of  gravity.

Hence    F=  —  ;
r

and   therefore,    neglecting  the   squares    &c.  of  very  small

dV    dV    dV
quantities,  F,  and  consequently  the  attractions  —  ,  —  ,  —  ,

are  the  same  as  if  the  whole  mass  of  the  Sun  were  collected
into  its  center  of  gravity.

77-  Now  take  the  principal  axes  of  the  Earth,  passing
through  its  center  of  gravity,  as  the  axes  of  co-ordinates,
the  polar  axis  being  that  of  %  ;  let  aoy%  be  the  co-ordinates
of  any  element  \$m  of  the  Earth,  and  xyss  the  co-ordi-
nates of  the  Sun,  supposed  to  be  condensed  into  his  center
of  gravity  ;  then,  m  being  the  mass  of  the  Sun,  his  at-
tractions on  £m,  parallel  to  the  axes  of  x  and  y,  will  be

andF=

71

and  hence  the  quantity   N,  (see  the  equations  (A')  of  the
last  Chapter)  which  equals  2\$w  (Yx  —  Xy)^  will  become

(y'-y)*-(*'-«>)y          \

*'-*m^)2+(*'-*)i*  '"I

\yoe-aiy)  [r'2-2(a!X+yy  +  **')  +  r2]"!  \  ,

r  and  r  being  the  distances  of  \$m  and  \$m  from  the  origin,
and  r  being  therefore  very  large  compared  with  a?,  y\  %9
or  r  ;  hence,  expanding  and  neglecting  the  squares  of  very
small  quantities,  we  have

or,   since   the  origin    is  center  of   gravity,    and    therefore
2^ma?,  S^my,  23m*,  each   zero,  and  since  the  axes  are
principal  axes,   and    therefore
each  zero,  we  have

and  ^f  being  the  same  as  in  the  last  Chapter.

Similarly,

8m'  *V

-  Q,

r

Sm'y'ss'

(C-B).

r'5
Hence  the  equations  (-4')j  in  the  last  Chapter,  become

A        +  (C  -  B)  W2»3  =  (C  -  K)

72

78.  These  equations  simplify  very  much  in  the  case
of  the  Earth,  for  the  polar  axis  being  that  of  *,  the
moments  of  inertia  round  all  axes  in  the  plain  of  xy  will
be  the  same,  since  the  Earth  is  symmetrical  with  respect
to  the  polar  axis  ;  hence  B  =  A,  and  the  last  equation  will
become

and  therefore  o>3  =  constant  =  n  suppose.

That  is  to  say,  by  (71),  the  angular  velocity  of  the  plane
of  the  equator  round  the  fixed  axis  with  which  the  polar
axis  coincides  at  any  time,  is  a  constant  quantity.

Now  if  ri  be  the  Sun's  mean  angular  velocity  relative  to
the  Earth,

n'2=  —  —  ,    m  being  the  Earth's  mass,

=  —  ,  very  nearly,  m  being  very  small   com-

pared with  TO';

C  —  A
hence,  and  putting  ft  for  —  -  —  ,   the  other   two  equations

become

79-  Now  in  the  figure  (page  66),  let  S  be  the  point
where  a  line  drawn  from  the  center  of  the  Earth  to  the  Sun
meets  the  fixed  sphere  mentioned  in  page  66;  draw  the
great  circle  SZ,  then  SZ  will  be  the  Sun's  north  polar  dis-
tance, which  we  shall  denote  by  A,  and  SZX  will  be  the
Sun's  hour-angle  relative  to  the  meridian  plane  ZX,  which
we  shall  denote  by  h :  it  is  evident  that  A  and  h  are  the
polar  co-ordinates  of  the  Sun,  and  therefore  we  have

73

x  =  r  sin  A  cos  A,
y  =  r  sin  A  sin  A,

z'  =  r  cos  A.

hence   -r-  =  sin  A  cos  A  sin  A,
ra

—  '—  =  sin  A  cos  A  cos  h  ;
r  a

and  hence  our  equations  become

—  +  w/3  wa  =  3w'2/3  sin  A  cos  A  sin  A,

2  -  w/3o>i=  —  3n'2fi  sin  A  cos  A  cos  A.
ct£

We  shall  take  a  year  as  the  unit  of  time,  and  hence  n',

2ir

which  =  -     —  ,   will  become  27r,  and  n  will  be  about  365,
a  year

also   we  shall  shew   that  /3  is  about  =  —  .

330

Now  h  varies  in  consequence  of  the  Earth's  diurnal
rotation,  and  also  in  consequence  of  the  Sun's  motion  and
the  motion  of  the  polar  axis  ;  but  the  part  of  its  variation
due  to  the  former  cause  is  very  much  greater  than  that  due
to  the  latter  causes,  hence  we  may  put

dh

where  \$n  is  a  small  quantity  compared  with  n,  depending
on  the  motions  of  the  Sun  and  polar  axis,  we  put  -  n  because
h  decreases  with  the  time.

Hence  in  the  first  of  our  equations,  putting
d<a\       dh)\  dh      dto,  ,  *

it  becomes

da,,  nfi  3n'*fi

e»>2  =  —    — s—  sin  A  cos  A  sin  n  ;

dh       -  n  4-  \$n          -  n  +
10

74
or,  neglecting  \$n  compared  with  w,

— -    -  /3ft>2=  —  Sra'*  —  sin  A  cos  A  sin  h.
ah  n

Now  the  second  member  of  this  equation  is  very  much
smaller  than  the  first,  on  account  of  being  multiplied  by

- ,  hence  in  it  we  may,  when  integrating,  consider  the
n

periodical  quantity  A  as  invariable,  since  it  varies  very
slowly  compared  with  h ;  and  therefore  putting,  for  brevity,

3n'2ft    .

—  sin  A  cos  A  =  y,
n

our  equation  becomes

^ -£«,••  -y  sin  h...  (1),

where  y  is  a  very  small  quantity  which  may  be  considered
invariable  in  integrating.

In  the  same  way  we  shall  have

-~-  +
ah

80.      Now  differentiating  (l),  and  adding  (2)  multiplied
by  ft  to  it,  we  have

_J2

-jj£  +  ft2u,}  =  (-  y  +  7/3)  cos  h

=  -  y  cos  h,
neglecting  ft  compared  with  unity.

The  integral  of  this  equation  will  be

o>i  =  A  cos  (fih  +  B)  +  C  cos  h,

where  A  and  B  are  arbitrary  constants,  and  C  a  constant  to
be  determined  by  substitution.  Now  A  and  B,  since  they
depend  only  on  the  initial  circumstances  of  the  motion,  are
independent  of  the  Sun's  action ;  and,  as  it  is  our  object  to

75

determine  the  effect  of  that  action  alone,  we  shall  omit  the
term

Acos(fih+  B),

which  does  not  depend  on  it,  and  we  shall  have  simply

(Dl  =  C  cos  h,

so  far  as  the  Sun's  action  is  concerned ;   substituting  this
value  in  the  equation,  we  find

C(-i  +  /r>  =  -7,
or  C  =  y,  neglecting  ft2 ;   hence

fcjj  =  y  cos  h.

In  like  manner  we  shall  have,  differentiating  the  equa-
tion (2),  and  subtracting  (1)  multiplied  by  /3  from  it,

d2a)2       „„  .

— —  +  /52ft>2  =  -  7  sin  h  ;
an

and  therefore,  as  before,

ft>2  =  y  sin  h.

81.  Having  thus  determined  &>!  and  ft>25  we  shall  sub-
stitute their  values  in  the  equations  (C)  (last  Chapter),  in
order  to  determine  9  and  0,  and  so  find  the  position  and
motion  of  the  pole ;  we  have  then,  substituting  the  values  of
o>i  and  ft>2  just  obtained  in  the  two  first  of  the  equations  (C),

dvl/

— ~  sin  9  =  y  sin  h  sin  0  -  y  cos  h  cos  <p

=  -  y  cos  (h  +  0),

dO

and   —  =  y  sin  h  cos  (f>  +  y  cos  h  sin  0

=  y  sin  (h  +  0).

Now  in  the  figure  (page  66),  the  angle  SZX  is  A,  and
the  angle  XZC  is  0,  hence  the  angle  *S*ZC  is  (f)  +  h;  there-
fore if  we  take  the  point  A  to  be  the  pole  of  the  ecliptic
(which  we  may  do  since  its  position  is  arbitrary),  it  is  evi-
dent that  ^  +  h  or  SZC  will  be  the  Sun's  right  ascension

76

—  90°;    hence,  if  a  be  the   Sun's  right  ascension,  we  shall
have

\$  +  h  =  a  -  90°,

and  hence  our  equations  become

d^    .

— —  sin  9  =  —  y  sin  a,

dO

-  =  -7cosa;

or,  putting  for  y  its  value,

d^   .  3n'2Q    .

— -  sin  6  = —  sin  A  cos  A  sin  a,

at  n

d9          3riz\$   .

—  = sin  A  cos  A  cos  a.

dt  n

82.  Now  if  /  be  the  Sun's  longitude,  /,  a,  and  90°  -  A
are  the  sides  of  a  right-angled  spherical  triangle,  the  right
angle  being  opposite  Z,  and  0  (the  obliquity  of  the  ecliptic)
being  the  angle  opposite  90°  —  A ;  hence  since,  by  Napier's
rules,

cos  A  =  sin  6  sin  I (1),

cos  Z  =  sin  A  cos  a (2),

and  sin  a  =  cot  A  cot  9 (3),

we  have

sin  A  cos  A  sin  a  =  cos2  A  cot  0,  by  (3),

=  sin  0  cos  9  sin2  Z,  by  (l),
and  sin  A  cos  A  cos  a  =  cos  Z  cos  A,  by  (2),

=  sin  9  sin  I  cos  Z,  by  (l)  ;
hence  our  equations  become

<ty          3n'20
I_  = r  cos  0  sm2  /,

dt  n

d9  3n'*P    .

a-    _    _ L-  SJn  0  Cos  /.

dt  n

77

83.     Now  —  being  very  small,  we  may  in  integrating
n

these  equations  consider  &  invariable  in  the  terms  multiplied

by  —  ;  and  we  may  also,  for  the  same  reason,  consider  I  to

n  dl

vary  uniformly,  and  therefore  put  —  =  n  ;  hence,  putting

d\/      d\,  dl      d

d9      dO    ,

Tt=-din>

and    sin2  /  =  1  (l  -  cos  2  Z),
our  equations  become

S—  £*

hence,  integrating

^  +  C  =  -  -  -  y3  .  cos  6  (I  -  J  sin  2/),

Q  +  C"  =  3  -  3  sin  0  sin  /.
n  ^

In  the  second  members  of  these  equations  0  may  be
considered  as  the  mean  obliquity  of  the  ecliptic,  and  may
therefore  be  determined  by  astronomical  observations;  and

—  ,  the  ratio  of  a  year  to  a  day,  may  be  similarly  deter-
n

mined  ;  and  thus  we  may  put  our  equations  in  the  form

9  +  C  =  gfi  sin  /,

where  e',  f  ',  g',  are  numerical  quantities  got  from  obser-
vation.

These  equations  determine  the  effect  of  the  Sun's  at-
traction on  the  motion  of  the  Earth  round  its  center  of
gravity.

78

84.  To  determine  the  effect  of  the  Moon  on  the  Earth's
motion  round  its  center  of  gravity,  we  may  proceed  in  ex-
actly a  similar  way,  merely  supposing  the  symbols  which
before  referred  to  the  Sun  now  refer  to  the  Moon,  and  alter-
ing them  accordingly,  as  follows  :

Let  my,  w/5  and  rt  be  the  mass,  mean  angular  velocity,
and  distance  of  the  Moon,  then  as  in  Art.  (78),  we  shall
have

here  m  instead  of  being  much  smaller  than  m  t  as  before,  is
much  larger  than  it,  and  therefore  cannot  be  neglected
as  before,

let  X  =  —  ,  then

n

m

,

and  therefore  —  '  =  -  —  ;
r,  3      X  +  1

hence,  in  changing  our  equations  so  as  to  refer  to  the  Moon,

M*

we  must  in   substituting  for  n2  put  —  '—    instead  of   nf

for  it.

Hence,  if  we  put  dashes  under  the  letters  to  denote
that  they  refer  to  the  Moon,  we  shall  have  for  the  Moon's
effect  on  the  Earth's  motion  round  its  center  of  gravity,

s  '  /  -     sin  '"'

Of  course  the  point  A  in  the  figure,  (page  66),  is  now
supposed  to  be  the  pole  of  the  Moon's  orbit,  and  not  the
pole  of  the  ecliptic.

79

85.  In  these  expressions  the  coefficients  are  much
smaller  than  before;  so  small  that  the  periodical  quantities
multiplied  by  them,  viz.  sin  2ntt  and  sin  nt^  which  go
through  all  their  values  in  half  a  month,  and  a  month
respectively,  may  be  neglected,  this  will  give

and

and  therefore  ---   =  -  -  •—-    —  cos  9  ,
dt  2n  (X  +  1)

,  dB

and  —  '  =  0,
dt

Which  equations  prove,  that  the  effect  of  the  Moon's
attraction  (omitting  very  small  periodical  quantities  of  short
period)  is  to  produce  a  motion  of  the  pole  of  the  Earth  per-
pendicular to  the  great  circle  AZ,  i.e.,  the  great  circle
joining  the  pole  of  the  Earth  and  the  pole  of  the  Moon's
orbit  ;  and  the  velocity  with  which  this  motion  takes  place

(i.e.  sin  0,),  is

cos  9  sin

We  shall  resolve  this  velocity  along  and  perpendicular
to  the  great  circle,  joining  the  pole  of  the  Earth  and  the
pole  of  the  ecliptic,  in  order  to  get  our  quantities  measured
in  the  same  way  as  before  in  the  case  of  the  Sun,  and  so  de-
termine the  variations  (due  to  the  Moon's  action),  of  the
angles  \js  and  0,  which  refer  to  the  pole  of  the  ecliptic.

86.  Let  i  be  the  inclination  of  the  Moon's  orbit  to  the
plane  of  the  ecliptic ;  then  it  is  evident  that  t,  0,  and  9t  form
the  sides  of  a  spherical  triangle.

80

Let  cr  be  the  angle  made  by  6  and  0/?  then  resolving  the
velocity

3«2/3

—  cos  9  sm  6  ,
2w  (X  +  1)

(which  acts  perpendicularly  to  0y),  along  and  perpendicular
to  9  we  find  for  the  resolved  parts,

--  7-*  -  -  cos  9  sin  0  cos  a   perpendicular  to  0,
2n  (\  +  1)

and  --  -r-*  -  -  cos  9f  sin  9t  sin  cr    along  0,
2  w  (A.  +  1}

and   hence,  since   —  sin  9  and  —  are  these  velocities,  we
at  at

have  the  following  equations  to  determine  the  effect  of  the
Moon's  action  on  \!/  and  9,  viz.

d\js  3nffl       cos  Qt  sin  9f  cos  cr

1/7          2n(\+  1)  sin  0

d0  3w//3

—  =  --  —  cos  9  sin  0  sin  cr.

dt

Now  in  the  triangle  of  which  t,  0,  and  0/  are  the  sides,
o-  is  the  angle  opposite  i,  and  if  &  be  the  longitude  of  the
Moon's  node,  it  is  evident  that  &  is  the  angle  opposite  Bt  ;
hence  we  have

cos  t  =  cos  9  cos  0/  +  sin  0  sin  0y  cos  cr,  ...  (l)
cos  Qt  =  cos  i  cos  0  +  sin  i  sin  0  cos  &  ,  ...  (2)
sin  0.  sin  t

(3)

am    56  alu  O"

(•n«  rt      sin  A     r-i\<   rr

hence

sin0

COS  t  -  COS  0  COS  0,

=  (cosi  cosfl  +  siru  sin0cos&  )

81

cos  t  sin2  0 — sin  t  cos0  sin  9  cos  &

sin2  9

(sin20-cos20)

cos2i  cos  0+sm  t  cos  i  —    —  —  -  -  -  cos  &  -sm2t  cos  9  cos2  &
sin  9

—  :  —  —
sin  9

and  also

cos  9,  sin  9  1  sin  a-

=  (cos  i  cos  9  +  sin  t  sin  9  cos  &  )  sin  t  sin  £
=  ^  sin  2t  cos  9  sin  &  +  ^  sin8  1  sin  0  sin  2  &  .

In  these  expressions  Q>  alone  may  be  considered  as  va-
riable, all  the  other  quantities  varying  very  slowly,  and
within  very  small  limits;  also,  since  these  expressions  are
to  be  multiplied  by  a  very  small  coefficient,  we  may,  in  the
periodical  terms,  neglect  sin2t,  since  t  is  not  much  more
than  5°.

Also,  if  v  be  the  mean  angular  retrograde  velocity  of
the  Moon's  nodes,  we  may  put

and  hence  we  shall  have  as  in  former  cases

3w2/3  A  sin2*  1

'  cos0{cos-<  -----  sin  2  1  cot  29  cos  Q  },

i/(\+l)  1  2  j

cos0sm2t  sin  &  ;

—  -      --
d&      2ni/(\

d9

and  hence,  integrating,

9  +  C'  =  -  -  —  /  ^    ^  cos0  sin2t  cos  &  ;

11

82

or,  as  before,  we  may  put  these  equations  in  the  form

3  3

^tc-^  —  «-/

A  +  1

When   e^  ft,  and  g    are  numerical  quantities,  got  by  ob-
servation.

Thus  we  have  determined  the  effect  of  the  Moon's
attraction  on  the  motion  of  the  Earth,  round  its  center
of  gravity.

87-  The  effects  of  the  other  planetary  bodies  are  very
small  indeed,  and  we  shall  neglect  them  ;  hence,  adding
together  the  effects  of  the  Sun  and  Moon,  we  find  for  the
whole  motion  of  the  Earth  round  its  center  of  gravity,  so
far  as  it  is  affected  by  external  attractions,

>/,  +  C  =  -  /3  (el  -  ~^~  Q  )  +  /'/3  sin  2  /

A.  +  1

0+  C'  =    's\i

/
A  +  I

The   first   term   of   the   expression  for  >//  +  C    is    non-
periodical,  and  its  rate  of  variation  is

or    -(3(e'n'  +  ^-_

which  being  constant  and  negative  represents  a  uniform
retrograde  motion  of  the  pole  in  longitude :  it  gives  rise  to
what  is  called  the  precession  of  the  equinoxes,  because,  in
consequence  of  it,  the  first  point  of  Aries  moves  constantly
backwards,  and  therefore  the  equinox  occurs  sooner  than  it
otherwise  would  every  year.

83

Observation  shews  that  this  retrograde  motion  of  the
pole  in  longitude  is  about  50".l  per  year;  hence  we  ought
to  have

The  other  terras  of  \|/  +  C  and  0  +  C'  are  periodical,
depending  on  the  longitude  of  the  Sun  and  of  the  Moon's
nodes;  they  are  called  the  solar  and  lunar  nutations.  Ob-
servation shews  that  the  coefficient  of  sin  &  is  about  18",
and  that  of  cos  Q  about  9".5,  the  coefficients  of  the  other
terms  are  much  smaller  ;  hence  we  ought  to  have

and  9».6-«r,r--  ............  (3).

A  +  1

88.  Since  e  ,  //5  and  g  are  known  numerical  quan-
tities, it  is  evident,  that  from  (l)  combined  with  (2)  or  (3),
we  may  eliminate  )3  and  find  X  ;  the  result  is

thus  by  observation  on  precession  and  nutation  we  may
determine  X,  which  is  the  ratio  of  the  Earth's  mass  to  that
of  the  Moon.

By  the  same  equations  we  may  determine  /3,  the  re-
sult is

/3  =  about  .00319,  or  -  .
*oO

89-  Now  /3  may  be  also  calculated  by  integration,  if
we  know  the  law  of  arrangement  of  the  Earth's  mass  ;  we
shall  calculate  /3,  assuming  the  results  arrived  at  in  Chap,  in,
and  if  we  find  that  the  value  of  /3  thus  obtained  coincides
with  that  just  determined  by  observation,  it  is  evident  that
we  shall  have  an  additional  proof  of  the  correctness  of  our
hypotheses  in  Chap.  in.

84

90.     We  have  evidently

»-»)      2Sm  (*»-«»)

'  S1

#,  y,  #  being  the  co-ordinates  of  any  particle  (\$m)  of  the
Earth,  the  polar  axis  being  the  axis  of  ss  ;  or  using  the  polar
co-ordinates  as  before,

Now  putting  ^  +  ^  cos  20  for  cos20,  and  integrating
relatively  to  0,  observing  that  r  does  not  contain  0,  the
numerator  of  )3  becomes

which,  putting

r'=«5{l   +5e(J  -!.")},

and  observing  the  property  of  Laplace^s  coefficients  in  Art.
21,  becomes

9      9      -

8

~  45'

hence  our  integral  becomes

Sir    ra    d(o5e)    .
—   /    p  --  da.

15  Jn

da

85
In  like  manner  the  denominator  of  /3  becomes

TT    f     r«,    d(O/4       /i         \1
=  -   /      /     p—  :  —  {  --     --  M     V
5  J-\  Jo    r    da     (3        \3          )  I

Since,  on  account  of  the  smallness  of  /3,  we  may  neglect  e
in  its  denominator.

Hence,  observing  the  property  in  Art.  21,  the  denomi
nator  of  /3  becomes

STT    /•<•!     rf(a5)  STT   /•«.

—  /     p  —  —  aa    which  =  —  /     pa*da.

15  Jo    *     da  3  A    ^

hence  we  have

5

/     p 1

1          <J  Q  (t  (t

r^

I     pa*da

«/o

Now  the  equation  for  determining  e  in  Art.  53,  gives

e,    /•"'     .  1      ra'    d(a5e)  a>zaj

--  ^  /    pa'da  +  —  -  /    p—     —  da  +  --  ^  =  0,
a,J0  r  *«/•'•  ^a  STT

r">  w^o/

a  e    /     pa^da  --  '-

0         '    Vt    r  STT

hence    p  =  -

rpa*  da
.    '

a~  I     pa?  da

'   Jo      '

I     pa*da

putting

m  being,  as  before,  the  ratio  of  the  centrifugal  force  to
gravity  at  the  equator.  We  can  go  no  farther  in  calculating
/3  without  knowing  the  law  of  density ;  hence,  taking  the
law  already  assumed,  and  substituting  in  the  integrals  and
performing  the  integrations,  we  shall  find

/3  =  about  .0031359,   or   .

330

91.  Hence  this  value  of  /3  coincides  with  that  got  from
observation,  in  Art.  88,  and  we  have  therefore  an  additional
proof  of  the  hypothesis  of  the  Earth's  fluidity,  or  rather,
of  the  assumed  law  of  density ;  for  since  this  result  cannot
be  obtained  without  assuming  the  law  of  density,  it  is  not  of
much  value  in  proving  the  hypothesis  of  original  fluidity ;
but  we  may  consider  that  hypothesis  as  well  established  by
previous  results,  and  then  the  coincidence  of  the  values  of  /3
will  go  to  proving  the  probability  of  the  assumed  law  of
density.

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TISE on  the  BOOK  of  COMMON  PRAYER.  By  the
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Cambridge,  and  Examining  Chaplain  to  the  Lord  Bishop
of  London.  Preparing.

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M.A.,  Rector  of  Buckland,  Herts.,  and  late  Fellow  of  King's
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derived  from  the  Writings  of  the  Older  Divines.  By  the
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Eucharist  Considered,  in  Reply  to  Dr.  WISEMAN'S  Argument,
from  Scripture.  By  THOMAS  TURTON,  D.D.,  some  time
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and  Dean  of  Peterborough,  NOW  BISHOP  OF  ELY.  A  New
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A  TRANSLATION  of  the  EPISTLES  of  OLE-

MENT  of  Rome,  Polycarp,  and  Ignatius ;  and  of  the
Apologies  of  Justin  Martyr  and  Tertullian :  with  an  Intro-
duction and  Brief  Notes  illustrative  of  the  Ecclesiastical
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rine's Hall.  New  Edition.  8vo.  12a.

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England,  from  its  compilation  to  the  last  revision ;  together
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14  Theology.

The  GOSPEL  according  to  ST.  MATTHEW,  and

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translated  into  English  from  the  Greek,  with  original  Notes,
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Gonville  and  Caius  College,  and  Minister  of  St.  Edward's,
Cambridge.  Price  6rf.  sewed;  8d.  stiff  wrappers.

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ECCLESI^E  ANGLICANS  VINDEX  CATHO-

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ROMA    RUIT:    The   Pillars    of    Rome    Broken.

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ditional matter,  by  C.  HARDWICK,  M.A.,  Fellow  of  St.
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John  Deighton,

Tlieology.  15

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The   DOCTRINE    of   the    GREEK    ARTICLE

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THE  THIRTY-NINE  ARTICLES,  TeJlimonies

and  Authorities,  Divine  and  Human,  in  Confirmation  of.
Compiled  and  arranged  for  the  use  of  Students.  By  the
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LEXICON  to  the  NEW  TESTAMENT,  a  Greek

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HORJE    HEBRAIC^E.     Critical  Observations   on

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Cambridge.  Svo.  8s.  <od.

to  Dr.  Turton's  Roman  Catholic  Doctrine  of  the  Eucharist
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Explaining  the  New  Testament  by  the  early  Opinions  of
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Cambridge.

CLASSICS,

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John  Deighton,

Classics.  X9

GEMS  of  LATIN  POETRY.     With  Translations,

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great  Oyer  of  Poisoning,  &c.  &c.  8vo.  12s.

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Fellow  and  Tutor  of  Trinity  College,  Cambridge.

8vo.  8s.  6d.

riTEPIAHS  KATA  AHMO20ENOTS.  The  Ora-
tion of  Hyperides  against  Demosthenes,  respecting  the
Treasure  of  Harpalus.  The  Fragments  of  the  Greek  Text,
now  first  Edited  from  the  Facsimile  of  the  MS.  discovered
at  Egyptian  Thebes  in  1847;  together  with  other  Fragments
of  the  same  Oration  cited  in  Ancient  Writers.  With  a  Pre-
liminary Dissertation  and  Notes,  and  a  Facsimile  of  a  por-
tion of  the  MS.  By  C.  BABINGTON,  M.A.,  Fellow  of
St.  John's  College,  Cambridge.  4  to.  6s.  6d.

ARUNDINES   CAMI.     Sive   Musarum   Cantabri-

giensium  Lusus  Canori ;  collegit  atque  edidit  H.  DRURY,
A.M.     Editio  quarta.  8vo.  12s.

DEMOSTHENES   DE   FALSA   LEGATIONE.

A  New  Edition,  with  a  careful  revision  of  the  Text,  Anno-
tatio  Critica,  English  Explanatory  Notes,  Philological  and
Historical,  and  Appendices.  By  R.  SHILLETO,  M.A.,
Trinity  College,  Cambridge.  8vo.  10s.  Qd.

DEMOSTHENES,  Translation  of  Select  Speeches

of,   with  Notes.      By   C.  R.   KENNEDY,   M.A.,   Trinity
College,  Cambridge.  12mo.  9s.

VARRONIANUS.  A  Critical  and  Historical  In-
troduction to  the  Philological  Study  of  the  Latin  Language.
By  the  Rev.  J.  W.  DONALDSON,  D.D.,  Head-Master  of
Bury  School,  and  formerly  Fellow  of  Trinity  College,  Cam-
bridge.

This  New  Edition,  which  has  been  in  preparation  for  several  years,  will
he  carefully  revised,  and  will  be  expanded  so  as  to  contain  a  complete
account  of  the  Ethnography  of  ancient  Italy,  and  a  full  investigation
of  all  the  most  difficult  questions  in  Latin  Grammar  and  Etymology,

Cambridge.

»2o  Classics  \

fidem  Manuscriptorum  emendatse  et  brevibus  Notis  emen-
dationum  potissimum  rationes  reddentibus  instructs.  Ediclit
R.  PORSON,  A.M.,  &c.,  recensuit  suasque  notulas  subjecit
J.  SCHOLEFIELD.  Editio  Tcrtia.  8vo.  10s.  6d.

TITUS   LIVIUS,    with   English  Notes,   Marginal

References,  and  various  Readings.  By  C.  W.  STOCKER,
D.D.,  late  Fellow  of  St.  John's  College,  Oxford.  Vols.  I.
and  II.,  in  4  Parts,  12s.  each.

GREEK  TRAGIC  SENARII,  Progressive  Exer-
cises in,  followed  by  a  Selection  from  the  Greek  Verses  of
Shrewsbury  School,  and  prefaced  by  a  short  Account  of  the
Iambic  Metre  and  Style  of  Greek  Tragedy.  By  the  Rev.
B.  H.  KENNEDY,  D.D.,  Prebendary  of  Lichfield,  and
Head-Master  of  Shrewsbury  School.  For  the  use  of  Schools
and  Private  Students.  Second  Edition,  altered  and  revised.

The   DIALOGUES   of  PLATO,   Schleiermacher's

Introductions    to.     Translated    from    the    German    by   "\V.
DOBSON,  A.M.,  Fellow  of  Trinity  College,  Cambridge.

8vo.  12s.  Gel.

M.  A.  PLAUTI  'AULULARIA.     Ad  fidem  Codi-

cum  qui  in  Bibliotheca  Musei  Britannici  exstant  aliorumque
nonnullorum  recensuit,  Notisque  et  Glossario  locuplete  in-
struxit  J.  HILDYARD,  A.M.,  Coll.  Christi  apud  Cantab.
Socius.  Editio  altera.  8vo.  7s.  6af.

M.  A.  PLAUTI  MEN^CHMEI.     Ad  fidem  Co-

dicum  qui  in  Bibliotheca  Musei  Britannici  exstant  aliorum-
que nonnullorum  recensuit,  Notisque  et  Glossario  locuplete
instruxit,  J.  HILDYARD,  A.M.,  etc.  Editio  altera.

Svo.  7s.  6d.

PHILOLOGICAL  MUSEUM.    2  vols.

8vo.  reduced  to  10s.
SOPHOCLES.     With  Notes  Critical  and  Explana-

tory,  adapted  to  the  use  of  Schools  and  Universities.  By
T.  MITCHELL,  A.M.,  late  Fellow  of  Sidney  Sussex  Col-
lege, Cambridge.  2  vols.  Svo.  11.  84.

Or  the  Plays  separately,  5s.  each.

John  Deighton,

Classics.  21

PROPERTIUS.    With  English  Notes.

By  F.  A.  PALEY,  Editor  of  .-Eschylus.  Preparing.

CORNELII  TACITI  OPERA.     Ad  Codices   an-

tiquissimos  exacta  et  emendata,  Commentario  critico  et
exegetico  illustrata.  Edidit  F.  HITTER,  Prof.  Bonnensis.
4  vols.  Svo.  II.  8s.

A  few  copies  printed  on  thick  Vellum  paper,  imp.  Svo.  41.  4«.

The  THEATRE  of  the  GREEKS.    A  series  of

Papers  relating  to  the  History  and  Criticism  of  the  Greek
Drama.  With  a  new  Introduction  and  other  alterations.
By  J.  W.  DONALDSON,  D.D.,  Head-Master  of  Bury
St.  Edmund's  Grammar  School.  Sixth  Edition.  8vo.  15*.

THEOCRITUS.     Codicum   Manuscriptorum  ope

recensuit  et  emeiidavit  C.  WORDSWORTH,  S.T.P.,  Schoke
Harroviensis  Magister,  nuper  Coll.  SS.  Trin.  Cant.  Socius
et  Academic  Orator  Publicus.  Svo.  IBs.  Qd.

A  few  copies  on  LARGE  PAPER.     4to.  II,  10s.

THUCYDIDES.    The  History  of  the  Peloponnesian

War :  illustrated  by  Maps  taken  entirely  from  actual  Surrevs.
With  Notes,  chiefly  Historical  and  Geographical.  By"T.
ARNOLD,  D.D.  Third  Edition.  3  rols.  Svo.  II.  10s.

THUCYDIDES.    The  History  of  the  Peloponnesian

War :  the  Text  of  ARNOLD,  with  his  Argument.  The
Indexes  now  first  adapted  to  his  Sections,  and  the  Greek
Index  greatly  enlarged.  By  the  Rev.  R.  P.  G.  TIDDEMAN.
M.A.,  of  Magdalene  College,  Oxford.  Svo.  12s,

Cambridge.

MISCELLANEOUS  WORKS.

LECTURES  on  the  HISTORY  of  MORAL

PHILOSOPHY  in  England.    By  W.  WHEWELL,  D.D.,

Master1  of  Trinity  CoUege,  Cambridge.  8vo.  8*.

THUCYDIDES    or  GROTE?      By  RICHAED

SHILLETO,  M.A.,  of  Trinity  College,  and  Classical  Lec-
turer of  King's  College,  Cambridge.  8vo.  2«.

A   FEW   REMARKS    on    a    Pamphlet    by    Mr.

SHILLETO,  entitled  "  THUCYDIDES  or  GROTE  ?"     2s.  6d.

The  HISTORY  of  the  JEWS  in  SPAIN,  from  the

time  of  their  Settlement  in  that  country  till  the  Commence-
ment of  the  present  Century.  Written,  and  illustrated  with
divers  extremely  scarce  Documents,  by  DON  ADOLPHO

Translated  by  the  Rev.  EDWARD  D.  G.  M.  KIRWAN,
M.A.,  Fellow  of  King's  CoUege,  Cambridge.  Crown  8vo.  6s.

The  QUEEN'S   COURT  MANUSCRIPT,    with

other  Ancient  Poems,  translated  out  of  the  original  Slavonic
into  English  Verse,  with  an  Introduction  and  Notes.  By
A.  H.  WRATISLAW,  M.A.,  Fellow  and  Tutor  of  Christ's
College,  Cambridge.  Fcap.  8vo.  4s.

On  the  EXPEDIENCY  of  ADMITTING  the  TES-
TIMONY of  Parties  to  Suits  in  the  New  County  Courts  and
in  the  Courts  of  Westminster  Hall.  To  which  are  appended,
General  Remarks  relative  to  the  New  County  Courts.  By
A.  AMOS,  Esq. ,  Judge  of  the  County  Courts  of  Marylebone,
Brompton,  and  Brentford,  Downing  Professor  of  Laws  in
the  University  of  Cambridge,  late  Member  of  the  Supreme
Council  of  India.  8vo.  3s.

The    LAWS    of    ENGLAND,     An   Introductory

Lecture  on,  delivered  in  Downing  College,  Cambridge,  Oc-
tober 23,  1850.  By  A.  AMOS,  Esq.,  Downing  Professor  of
the  Laws  of  England.  8vo.  1«.  6d.

John  Deighton,

Miscellaneous  Works.  23

HISTORY  of  ROME.     By T.  ARNOLD,  D.D.,  late

3  University  of
18s.;  Vol.  III.,

Regius  Professor  of  Modern  History  in  the  University  of
Oxford.     3  vols.  8vo.     Vol.  I.,  16*.;  Vol.  II.,

14s.

HISTORY  of  the  LATE  ROMAN  COMMON-
WEALTH, from  the  end  of  the  Second  Punic  War,  to  the
Death  of  Julius  Caesar,  and  of  the  Reign  of  Augustus  ;  "with
a  Life  of  Trajan.  By  T.  ARNOLD,  D.D.,  &c.  2  vols.

8vo-  II.  83.

Mr.  MACAULAY'S    Character  of  the  CLERGY

in  the  latter  part  of  the  Seventeenth  Century  Considered.
With  an  Appendix  on  his  Character  of  the  Gentry  as  given
in  his  History  of  England.  By  C.  BABINGTON,  M.A.,
Fellow  of  St.  John's  College,  Cambridge.  8vo.  4s.  6d.

CAMBRIDGE   UNIVERSITY    ALMANAC    for

the  Year  1852.  Embellished  with  a  Line  Engraving,  by
Mr.  E.  CHALLIS,  of  a  View  of  the  Interior  of  Trinity
College  Library,  from  a  Drawing  by  B.  RUDGE.  (Con-
tinued Annually.)  4s.  6d.

DESCRIPTIONS  of  the  BRITISH  PALEOZOIC

FOSSILS  added  by  Professor  Sedgwick  to  the  Woodwardian
Collection,  and  contained  in  the  Geological  Museum  of  the
University  of  Cambridge.  With  Figures  of  the  new  and
imperfectly  known  Species.  By  F.  M'COY,  Professor  of
Geology,  &c.,  Queens  College,  Belfast ;  Author  of  "  Car-
boniferous Limestone  Fossils  of  Ireland,"  '•  Synopsis  of  the
Silurian  Fossils  of  Ireland."  Parti.  4to.

logus  eorum  quos  ab  anno  1760  usque  ad  10m  Octr.  1846,
A.M.,  Coll.  Trin.  Socii  atque  Academise  Registrarii.

8vo.  10s.

MAKAMAT;    or  Rhetorical  Anecdotes  of  Hariri

of  Basra,  translated  from  the  Arabic  into  English  Prose  and
Verse,  and  illustrated  with  Annotations.  By  THEODORE
PRESTON,  M.A.,  Fellow  of  Trinity  College,  Cambridge.

Royal  8vo.  18s.;  large  paper,  II.  4s.

Cambridge.

24  Miscellaneous  Works.

ARCHITECTURAL    NOTES    ON    GERMAN

CHURCHES ;  with  Notes  written  during  an  Architectural
Tour  in  Picardy  and  Normandy.  By  W.  WHEWELL,
D.D.,  Master  of  Trinity  College,  Cambridge.  Third  Edition.
To  which  is  added,  Translation  of  Notes  on  Churches  of  the
Rhine,  by  M.  F.  De  LASSATJLX,  Architectural  Inspector
to  the  King  of  Prussia.  Plates.  8vo.  12s.

REMARKS    on  the    ARCHITECTURE    of   the

MIDDLE  AGES,  especially  in  Italy.  By  R.WILLIS,
M.A.,  Jacksonian  Professor  of  the  University  of  Cambridge.
Plates,  LAEGE  PAPER.  Royal  8vo.  il.  1*.

Views  of  the  Colleges  and  other  Public  Buildings

IN  THE  UNIVERSITY  OF   CAMBRIDGE,

Taken  expressly  for  the  UNIVERSITY  ALMANACK,    (measuring
abo-ut  17  inches  by  11  inches).

Cambridge  Antiquarian  Society's  Publications,

Nos.  I.  to  XV.     Demy  4to.,  sewed.

A     000426824     9

Sh'n'tlii  n-ilt  hs  PuUixlicd.  !>;!  the  wwr  Aitthnr :

MATHEMATICAL    TRACTS,

PART    II.

Containing,  among  other  subjects,  The  MATHEMATICAL
THEORY  of  ELECTRICITY,  briefly  considered,— The
LIBRATIONS  of  the  MOON,  ^-  The  THEORY  of
JUPITER'S  SATELLITES;  and  a  Short  ESSAY  on
TIDES.

```